**Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell**

Yongqiang Guo and Weiqiu Chen

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54799

## **1. Introduction**

24 Will-be-set-by-IN-TECH

[21] Ventura, P.,Hodé, J.M., Solal, M., Desbois, J. Ribbe, J., Numerical methods for SAW propagation characterization, Proc. of the IEEE Ultrasonics Symposium, pp175-186, 1998. [22] Hodé, J.M., Desbois, J., Original basic properties of the Green's functions of a semi-infinite piezoelectric substrate, Proc. of the IEEE Ultrasonics Symposium,

[23] Boyer, L., Étude des phénomènes de réflexion/réfraction dóndes planes acoustiques dans les milieux piézoélectriques, Thèse de lÚniversité de Paris VII an Acoustique

[24] Ribbe, J., On the coupling of integral equations and finite element/Fourier modes for the simulation of piezoelectric surface wave component, Phd Thesis, École Polytechnique,

[25] Berenger, J.P., Three-dimensional perfectly matched layer for the absorption of

[26] Laroche, T., Baida, F.I., Van Labeke, D., Three -dimensional finite-difference time domain study of enhanced second-harmonic generation at the end of a apertureless scanning

[27] Zheng, Y., Huang, X., Anisotropic Perfectly Matched Layers for Elastic Waves in Cartesian and Curvilinear Coordinates, 2002 MIT Earth Resources Laboratory Industry

electromagnetic waves, J. Comp. Phy. 127, 363–79, 1996

near-field optical microscope metal tip, josab 22, 1045–1051, 2005

pp131-136, 1999.

2002.

Physique, Paris, 1994

Consortium Report

The bulk acoustic wave (BAW) devices first emerged in 1920s and the surface acoustic wave (SAW) devices first appeared in 1960s (Royer & Dieulesaint, 2000). Since invented, these acoustic wave devices have been improved greatly in their performance and applications, along with significantly extended working parameters and application areas (Royer & Dieulesaint, 2000; Hashimoto, 2000). Nevertheless, in the last two decades, even more rigorous demands such as high operational frequency, high sensitivity, high reliability, multiple functionality, broad environment applicability, low attenuation and low cost, arise from the consumer, commercial and military applications. These demands challenge the conventional acoustic wave devices in which single crystalline piezoelectric materials are used as the wave medium. Therefore, the scheme of innovative acoustic wave devices utilizing piezoelectric multi-layered (stratified) structures was presented to cater for these demands. Fortunately, the successes of thin film deposition, etching and lithography technologies lead to the availability of piezoelectric multi-layered structures (Benetti et al., 2005). Recently, high-performance acoustic wave devices with multi-layered structures have been contrived and successfully fabricated (Kirsch et al., 2006; Benetti et al., 2008; Nakanishi et al., 2008; Brizoual et al., 2008). To further reduce the acoustic loss and enhance the quality factor of the multi-layered acoustic wave devices, Bragg Cell composed of many thin periodic alternate high- and low-impedance sublayers can be inserted between the propagation layer and the substrate. Efforts have been made on the fabrication of integrated piezoelectric multi-layered materials with Bragg Cell (Yoon & Park, 2000) and on the realization of superior multi-layered acoustic wave devices with Bragg Cell (Chung et al., 2008), especially aiming at the film bulk acoustic resonators (FBAR).

To ensure the well and stable performance of multi-layered acoustic wave devices, clear understanding of their operation status, especially the acoustic wave propagation behavior, is indispensable in the design process. Therefore, accurate and reliable modeling methods

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 267

acoustic wave devices working with various acoustic modes including Rayleigh modes, Love modes, Lamb modes, SH modes and bulk longitudinal/transversal modes, so as to improve their performances (Yoon & Park, 2000; Chung et al., 2008). Moreover, for acoustic wave devices working with a specific acoustic mode, other spurious modes inevitably exist. Therefore, for appropriately designing the multi-layered acoustic wave devices with Bragg Cell, modeling methods should be established by considering various wave modes and based on an integrated model, which reckoning on the propagation media, electrodes, Bragg Cell, support layer and substrate. In addition, for appropriately designing the Bragg Cell to improve the performance of multi-layered acoustic wave devices, the features and the mechanisms of frequency bands in the Bragg Cell should be studied. The influence of inserted Bragg Cell on

In this chapter, the wave behavior in the Bragg Cell and the design rules of a Bragg Cell are studied by taking SH wave mode as illustration and by using the Method of Reverberation-Ray Matrix (MRRM). The MRRM is also proposed for accurate analysis and design of multilayered acoustic wave devices with Bragg Cell, based on an integrated model involving the effects of electrodes, Bragg Cell, support layer and substrate on the working media. Firstly, the MRRM is extended to the analysis of SH wave dispersion characteristics of a ternary Bragg Cell, whose unit cell consisting of three isotropic layers. Based on the resultant closedform dispersion equations, the formation mechanisms of the SH wave frequency bands are revealed. The design rules of the Bragg Cell according to specific isolation requirements of SH waves are summarized. Secondly, the integrated model, which incorporates the effects of electrodes, Bragg Cell, support layer and substrate on the working piezoelectric media by modeling them as individual non-piezoelectric or piezoelectric layers, is proposed for accurately analyzing acoustic wave propagation in multilayered acoustic wave devices. The formulation of MRRM for the integrated multi-layered structures based on the state space formalism is derived, by which the propagation characteristics of waves can be investigated. In view of the achieved dispersion characteristics, the operating status of various acoustic wave devices can be decided. Thirdly, numerical examples are given to validate the proposed MRRM, to show the features and the formation of SH-wave bands in the Bragg Cell and to indicate the resonant characteristics of multi-layered acoustic wave devices. Finally, conclusions are drawn concerning the SH wave behavior in the Bragg Cell, the advantages of the integrated model and MRRM, and the resonant characteristics of multi-

acoustic wave propagation in the working layer should also be clearly revealed.

**2. The features and formation of SH-wave bands in the Bragg Cell** 

Consider an infinite periodic layered structure with each unit cell containing three isotropic elastic layers. A unit cell is depicted in Fig. 1, which can completely determine the band features of the infinite periodic layered structure by invoking the Floquet-Bloch principle (Mead, 1996). The surfaces and interfaces of the unit cell are denoted by numerals 1 to 4 from top to bottom, and the layers are represented by numerals 1 to 3 from top to bottom. Due to the isotropy of the layers, the in-plane wave motion is decoupled from the out-ofplane one. We limit our discussion to the out-of-plane (transverse) wave motion, i.e. only

layered acoustic wave devices.

the SH type mode is present.

are necessary. By far, three sorts of matrix methods, including the analytical methods based on continuous (distributed-parameter) models, numerical methods based on discrete models and analytical-numerical mixed methods, have been presented for analyzing multilayered acoustic wave devices. Analytical matrix methods, such as the transfer matrix method (TMM) (Lowe, 1995; Adler, 2000), the effective permittivity matrix method (Wu & Chen, 2002), the scattering matrix method (Pastureaud et al., 2002), and the recursive asymptotic stiffness matrix method (Wang and Rokhlin, 2002), usually give accurate results with low computational cost. However, some of these analytical methods are numerically instable. One reason is that both exponentially growing and decaying terms with respect to frequency and thickness are incorporated in a same matrix, and the other is that matrix inversion is involved in the formulation. For example, TMM ceases to be effective for cases of high frequency-thickness products. Tan (2007) compared most analytical methods in their mathematical algorithm, computational efficiency and numerical stability. Very recently, Guo et al. (Guo, 2008; Guo & Chen, 2008a, 2010; Guo et al., 2009) have presented a new version of the analytical method of reverberation-ray matrix (MRRM) formerly proposed by Pao et al. (Su et al., 2002; Pao et al, 2007), based on three-dimensional elasticity/ piezoelectricity (Ding & Chen, 2001), state-space formalism (Stroh, 1962) and plane wave expansion for the analysis of free waves in multi-layered anisotropic structures. The new formulation of MRRM deals with the exponentially growing and decaying terms separately and refrains from matrix inversion. It is a promising analytical matrix method, which bearing unconditionally numerical stability, for accurately modeling the multi-layered acoustic wave devices (Guo and Chen, 2008b). Numerical methods, including the finite difference method (FDM), the finite element method (FEM), the boundary element method (BEM) and the hybrid method of BEM/FEM (Makkonen, 2005), are powerful for modeling multi-layered acoustic wave devices with complex geometries and boundaries. However, they are less accurate and efficient, especially for high frequency analysis. The reason is that the wave media should be modeled by tremendous elements of small size to ensure computational convergence. Analytical-numerical mixed methods, such as the finite element method/boundary integral formulation (FEM/BIF) (Ballandras et al., 2004) and the finite element method/spectral domain analysis (FEM/SDA) (Hashimoto et al., 2009; Naumenko, 2010), are usually powerful for modeling both the small-sized accessories and the largedimensioned wave media with high accuracy. They seem to be promising as long as the uniformity of their formulation is improved (Hashimoto et al., 2009; Naumenko, 2010). Although some of these matrix methods are extendable to modeling the multi-layered acoustic wave devices with Bragg Cell, there are few investigations focused on this subject. Few studies have been reported on the effects of a Bragg Cell on wave propagation characteristics of multi-layered acoustic wave devices either. To the authors' knowledge, all existing references aimed at Bragg Cell in solidly mounted resonators. Zhang et al. (2006, 2008) and Marechal et al. (2008) studied both the resonant transmission in Bragg Cell and acoustic wave propagation in multi-layered bulk acoustic devices with Bragg Cell. Tajic et al., (2010) presented FEM combined with BEM and/or PML to simulate the solidly mounted BAW resonators with Bragg Cell. The formation mechanisms of the frequency bands in Bragg Cell are still an untouched topic. It should be pointed out that the Bragg Cell, as a kind of reliable wave guiding and isolating structure, is potential for utilizing in multi-layered acoustic wave devices working with various acoustic modes including Rayleigh modes, Love modes, Lamb modes, SH modes and bulk longitudinal/transversal modes, so as to improve their performances (Yoon & Park, 2000; Chung et al., 2008). Moreover, for acoustic wave devices working with a specific acoustic mode, other spurious modes inevitably exist. Therefore, for appropriately designing the multi-layered acoustic wave devices with Bragg Cell, modeling methods should be established by considering various wave modes and based on an integrated model, which reckoning on the propagation media, electrodes, Bragg Cell, support layer and substrate. In addition, for appropriately designing the Bragg Cell to improve the performance of multi-layered acoustic wave devices, the features and the mechanisms of frequency bands in the Bragg Cell should be studied. The influence of inserted Bragg Cell on acoustic wave propagation in the working layer should also be clearly revealed.

266 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

are necessary. By far, three sorts of matrix methods, including the analytical methods based on continuous (distributed-parameter) models, numerical methods based on discrete models and analytical-numerical mixed methods, have been presented for analyzing multilayered acoustic wave devices. Analytical matrix methods, such as the transfer matrix method (TMM) (Lowe, 1995; Adler, 2000), the effective permittivity matrix method (Wu & Chen, 2002), the scattering matrix method (Pastureaud et al., 2002), and the recursive asymptotic stiffness matrix method (Wang and Rokhlin, 2002), usually give accurate results with low computational cost. However, some of these analytical methods are numerically instable. One reason is that both exponentially growing and decaying terms with respect to frequency and thickness are incorporated in a same matrix, and the other is that matrix inversion is involved in the formulation. For example, TMM ceases to be effective for cases of high frequency-thickness products. Tan (2007) compared most analytical methods in their mathematical algorithm, computational efficiency and numerical stability. Very recently, Guo et al. (Guo, 2008; Guo & Chen, 2008a, 2010; Guo et al., 2009) have presented a new version of the analytical method of reverberation-ray matrix (MRRM) formerly proposed by Pao et al. (Su et al., 2002; Pao et al, 2007), based on three-dimensional elasticity/ piezoelectricity (Ding & Chen, 2001), state-space formalism (Stroh, 1962) and plane wave expansion for the analysis of free waves in multi-layered anisotropic structures. The new formulation of MRRM deals with the exponentially growing and decaying terms separately and refrains from matrix inversion. It is a promising analytical matrix method, which bearing unconditionally numerical stability, for accurately modeling the multi-layered acoustic wave devices (Guo and Chen, 2008b). Numerical methods, including the finite difference method (FDM), the finite element method (FEM), the boundary element method (BEM) and the hybrid method of BEM/FEM (Makkonen, 2005), are powerful for modeling multi-layered acoustic wave devices with complex geometries and boundaries. However, they are less accurate and efficient, especially for high frequency analysis. The reason is that the wave media should be modeled by tremendous elements of small size to ensure computational convergence. Analytical-numerical mixed methods, such as the finite element method/boundary integral formulation (FEM/BIF) (Ballandras et al., 2004) and the finite element method/spectral domain analysis (FEM/SDA) (Hashimoto et al., 2009; Naumenko, 2010), are usually powerful for modeling both the small-sized accessories and the largedimensioned wave media with high accuracy. They seem to be promising as long as the uniformity of their formulation is improved (Hashimoto et al., 2009; Naumenko, 2010). Although some of these matrix methods are extendable to modeling the multi-layered acoustic wave devices with Bragg Cell, there are few investigations focused on this subject. Few studies have been reported on the effects of a Bragg Cell on wave propagation characteristics of multi-layered acoustic wave devices either. To the authors' knowledge, all existing references aimed at Bragg Cell in solidly mounted resonators. Zhang et al. (2006, 2008) and Marechal et al. (2008) studied both the resonant transmission in Bragg Cell and acoustic wave propagation in multi-layered bulk acoustic devices with Bragg Cell. Tajic et al., (2010) presented FEM combined with BEM and/or PML to simulate the solidly mounted BAW resonators with Bragg Cell. The formation mechanisms of the frequency bands in Bragg Cell are still an untouched topic. It should be pointed out that the Bragg Cell, as a kind of reliable wave guiding and isolating structure, is potential for utilizing in multi-layered

In this chapter, the wave behavior in the Bragg Cell and the design rules of a Bragg Cell are studied by taking SH wave mode as illustration and by using the Method of Reverberation-Ray Matrix (MRRM). The MRRM is also proposed for accurate analysis and design of multilayered acoustic wave devices with Bragg Cell, based on an integrated model involving the effects of electrodes, Bragg Cell, support layer and substrate on the working media. Firstly, the MRRM is extended to the analysis of SH wave dispersion characteristics of a ternary Bragg Cell, whose unit cell consisting of three isotropic layers. Based on the resultant closedform dispersion equations, the formation mechanisms of the SH wave frequency bands are revealed. The design rules of the Bragg Cell according to specific isolation requirements of SH waves are summarized. Secondly, the integrated model, which incorporates the effects of electrodes, Bragg Cell, support layer and substrate on the working piezoelectric media by modeling them as individual non-piezoelectric or piezoelectric layers, is proposed for accurately analyzing acoustic wave propagation in multilayered acoustic wave devices. The formulation of MRRM for the integrated multi-layered structures based on the state space formalism is derived, by which the propagation characteristics of waves can be investigated. In view of the achieved dispersion characteristics, the operating status of various acoustic wave devices can be decided. Thirdly, numerical examples are given to validate the proposed MRRM, to show the features and the formation of SH-wave bands in the Bragg Cell and to indicate the resonant characteristics of multi-layered acoustic wave devices. Finally, conclusions are drawn concerning the SH wave behavior in the Bragg Cell, the advantages of the integrated model and MRRM, and the resonant characteristics of multilayered acoustic wave devices.

## **2. The features and formation of SH-wave bands in the Bragg Cell**

Consider an infinite periodic layered structure with each unit cell containing three isotropic elastic layers. A unit cell is depicted in Fig. 1, which can completely determine the band features of the infinite periodic layered structure by invoking the Floquet-Bloch principle (Mead, 1996). The surfaces and interfaces of the unit cell are denoted by numerals 1 to 4 from top to bottom, and the layers are represented by numerals 1 to 3 from top to bottom. Due to the isotropy of the layers, the in-plane wave motion is decoupled from the out-ofplane one. We limit our discussion to the out-of-plane (transverse) wave motion, i.e. only the SH type mode is present.

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 269

 

 

(2)

 

are respectively the *z* -direction shear

*G* is the shear stress coefficient, constants *G* and

ˆ ˆ () ( ) *JK JK KJ JK JK vz vh z* , (3)

and *z* , one obtains the phase relation of layer *j* (*JK* or *KJ*)

can always be chosen to satisfy Re[ i ] 0 *sj j*

*JK JK KJ <sup>s</sup> <sup>h</sup> JK KJ JK a d Pk d* (5)

*<sup>j</sup> PPP* are the wavenumber along *z*

*z hz* . (4)

being the circular frequency and

 (1)

in any

*<sup>z</sup>* and

*h* , so that

According to the elastodynamics of linear isotropic media (Eringen & Suhubi, 1975), the

constituent layer *JK* (or *j*) in its pertaining coordinates (,,) *JK JK JK xyz* can be expressed as

3 3 ( , , ) ( )e ( e e )e , <sup>ˆ</sup> *s s t kx z z t kx vxzt vz a d*

 

*k* ) domain (i.e. the transformed quantities), i 1 is the imaginary unit, *<sup>s</sup>*

 are respectively the shear modulus and the mass density. It is clearly seen that the terms with 3 *a* and 3 *d* at the right-hand sides of Eqs. (1) and (2) signify backward and forward traveling waves along the thickness coordinate, i.e. the arriving and departing waves relative to the surface *J*, with 3 *a* and 3 *d* being the corresponding undetermined wave amplitudes.

First, we consider the spectral equations within the layers. The transformed displacement *v*ˆ

(,,) *JK JK JK xyz* should be compatible with that expressed in the other coordinate system (,,) *KJ KJ KJ xyz* , due to the uniqueness of the physical essence. Referring to the sign

ˆ ˆ () ( ) *JK JK KJ JK JK*

<sup>3</sup> 33 3 e (, )

direction, the thickness and the phase coefficient of layer *j* (*JK* or *KJ*), respectively. It is noted

*JK KJ*

 

Substituting Eq. (1) into Eq. (3) and Eq. (2) into Eq. (4), and noticing the functions <sup>i</sup> e *<sup>s</sup>*

*zy zy*

i

*<sup>j</sup> hhh* and 333

no exponentially growing function is included in the phase relation.

, *JK KJ*

) at an arbitrary plane *JK z* of any layer *j* expressed in one coordinate system

33 3 ( , , ) ( )e ( e e )e , <sup>ˆ</sup> *s s t kx z z t kx*

i( ) i i i( )

 

i( ) i i i( )

*z* signify the corresponding quantities in the frequency-wavenumber

plane wave solutions to the out-of-plane displacement *v* and shear stress *zy*

*zy zy xzt z a d* 

*<sup>s</sup> k k* ) and / *s s k c*

 *s*

follows (with superscripts *JK* omitted)

*<sup>s</sup> k k* or 2 2

where *v z* ˆ( ) and ˆ ( ) *zy*

(

( 2 2

/ *<sup>s</sup> c G*

(stress ˆ

<sup>i</sup> e *<sup>s</sup>* 

where *JK KJ*

*zy* 

wavenumber and the total shear wavenumber with

the shear wave velocity, 3 i

convention of displacement (stress), we have

*<sup>z</sup>* are nonzero for finite *<sup>s</sup>*

*s s sj*

 

that the thickness wavenumber *sj*

 

**Figure 1.** The schematic of the unit cell of a periodic ternary layered structure and its description in the global coordinates

#### **2.1. SH wave dispersion characteristics of the Bragg Cell**

Within the framework of the method of reverberation-ray matrix (MRRM) (Su et al., 2002; Pao et al., 2007; Guo, 2008), constituent layers of the unit cell are individually described in the corresponding local dual coordinates. Fig. 2 depicts the local dual coordinates of a typical layer *j* ( *j* 1,2,3 ) with its top and bottom surfaces denoted respectively as *J* ( *j* ) and *K* ( *j*+1 ), and the SH wave amplitudes along the thickness in the typical layer *j* as the wavenumber along *X* is *k* for all of the constituent layers. Meanwhile, superscripts *JK* or *KJ* will be attached to physical variables of the typical layer *j* , which is also called as *JK* or *KJ* according to the related coordinates (,,) *JK JK JK xyz* or (,,) *KJ KJ KJ xyz* , to indicate the layer and its pertaining coordinate system. The displacement and stress are deemed to be positive as they are along the positive direction of the pertaining coordinates.

**Figure 2.** Description of a typical constituent layer *j* of the unit cell in local dual coordinates

According to the elastodynamics of linear isotropic media (Eringen & Suhubi, 1975), the plane wave solutions to the out-of-plane displacement *v* and shear stress *zy* in any constituent layer *JK* (or *j*) in its pertaining coordinates (,,) *JK JK JK xyz* can be expressed as follows (with superscripts *JK* omitted)

268 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

**2.1. SH wave dispersion characteristics of the Bragg Cell** 

*Y*

1

2

3

4

as they are along the positive direction of the pertaining coordinates.

*JK <sup>z</sup> JK <sup>y</sup>*

*<sup>Y</sup> <sup>Z</sup>*

*O*

*KJ x KJ y*

*K*

( 1) *j*

*J* ( )*j*

**Figure 2.** Description of a typical constituent layer *j* of the unit cell in local dual coordinates

*KJ z*

*JK x*

*X*

*j*

*j h*

1 *h*

1

*X*

1 1 *G* ,

2 2 *G* ,

3 3 *G* ,

2 *h*

2

3 *h*

3

3 *JK d*

3 *JK a*

*k*

3 *KJ a*

3 *KJ d*

global coordinates

**Figure 1.** The schematic of the unit cell of a periodic ternary layered structure and its description in the

*Z*

*O*

Within the framework of the method of reverberation-ray matrix (MRRM) (Su et al., 2002; Pao et al., 2007; Guo, 2008), constituent layers of the unit cell are individually described in the corresponding local dual coordinates. Fig. 2 depicts the local dual coordinates of a typical layer *j* ( *j* 1,2,3 ) with its top and bottom surfaces denoted respectively as *J* ( *j* ) and *K* ( *j*+1 ), and the SH wave amplitudes along the thickness in the typical layer *j* as the wavenumber along *X* is *k* for all of the constituent layers. Meanwhile, superscripts *JK* or *KJ* will be attached to physical variables of the typical layer *j* , which is also called as *JK* or *KJ* according to the related coordinates (,,) *JK JK JK xyz* or (,,) *KJ KJ KJ xyz* , to indicate the layer and its pertaining coordinate system. The displacement and stress are deemed to be positive

$$\upsilon(\mathbf{x}, z, t) = \hat{\upsilon}(z) \mathbf{e}^{\mathbf{i}(\alpha t - kx)} = (a\_3 \,\mathrm{e}^{\mathbf{i}\,\gamma\_s z} + d\_3 \,\mathrm{e}^{-\mathbf{i}\,\gamma\_s z}) \mathbf{e}^{\mathbf{i}(\alpha t - kx)},\tag{1}$$

$$\tau\_{zy}(\mathbf{x}, z, t) = \hat{\tau}\_{zy}(z) \mathbf{e}^{\mathrm{i}(\alpha t - k\mathbf{x})} = \zeta\_3(a\_3 \, \mathrm{e}^{\mathrm{i}\gamma\_s z} - d\_3 \, \mathrm{e}^{-\mathrm{i}\gamma\_s z}) \mathbf{e}^{\mathrm{i}(\alpha t - k\mathbf{x})},\tag{2}$$

where *v z* ˆ( ) and ˆ ( ) *zy z* signify the corresponding quantities in the frequency-wavenumber ( *k* ) domain (i.e. the transformed quantities), i 1 is the imaginary unit, *<sup>s</sup>* ( 2 2 *<sup>s</sup> k k* or 2 2 *<sup>s</sup> k k* ) and / *s s k c* are respectively the *z* -direction shear wavenumber and the total shear wavenumber with being the circular frequency and / *<sup>s</sup> c G* the shear wave velocity, 3 i *s G* is the shear stress coefficient, constants *G* and are respectively the shear modulus and the mass density. It is clearly seen that the terms with 3 *a* and 3 *d* at the right-hand sides of Eqs. (1) and (2) signify backward and forward traveling waves along the thickness coordinate, i.e. the arriving and departing waves relative to the surface *J*, with 3 *a* and 3 *d* being the corresponding undetermined wave amplitudes.

First, we consider the spectral equations within the layers. The transformed displacement *v*ˆ (stress ˆ *zy* ) at an arbitrary plane *JK z* of any layer *j* expressed in one coordinate system (,,) *JK JK JK xyz* should be compatible with that expressed in the other coordinate system (,,) *KJ KJ KJ xyz* , due to the uniqueness of the physical essence. Referring to the sign convention of displacement (stress), we have

$$
\partial^{\rm JK}(z^{\rm JK}) = \partial^{\rm KJ}(h^{\rm JK} - z^{\rm JK}) \,, \tag{3}
$$

$$
\hat{\pi}\_{zy}^{\rm IK}(z^{\rm IK}) = -\hat{\pi}\_{zy}^{\rm KJ}(h^{\rm IK} - z^{\rm IK}) \,. \tag{4}
$$

Substituting Eq. (1) into Eq. (3) and Eq. (2) into Eq. (4), and noticing the functions <sup>i</sup> e *<sup>s</sup> <sup>z</sup>* and <sup>i</sup> e *<sup>s</sup> <sup>z</sup>* are nonzero for finite *<sup>s</sup>* and *z* , one obtains the phase relation of layer *j* (*JK* or *KJ*)

$$d\_{\mathfrak{z}}^{\,^{\,\,\,\,}\,} = \mathfrak{e}^{-\,^{\,i}\gamma\_{\,\,\,}^{\,\,\,\,}\, h^{\,\,\,}} \, d\_{\mathfrak{z}}^{\,\,\,\,} = P\_{\mathfrak{z}}^{\,\,\,\,\,\,}(k, \,\boldsymbol{\alpha}) d\_{\mathfrak{z}}^{\,\,\,\,\,\,\,} \tag{5}$$

where *JK KJ s s sj* , *JK KJ <sup>j</sup> hhh* and 333 *JK KJ <sup>j</sup> PPP* are the wavenumber along *z* direction, the thickness and the phase coefficient of layer *j* (*JK* or *KJ*), respectively. It is noted that the thickness wavenumber *sj* can always be chosen to satisfy Re[ i ] 0 *sj j h* , so that no exponentially growing function is included in the phase relation.

Second, we consider the spectral equations at the interfaces between adjacent layers. The compatibility of displacements and equilibrium of stresses at the interfaces 2 and 3 are expressed as

$$
\hat{\boldsymbol{\upsilon}}^{21}(\mathbf{0}) = \hat{\boldsymbol{\upsilon}}^{23}(\mathbf{0}), \quad \hat{\boldsymbol{\tau}}^{21}\_{zy}(\mathbf{0}) + \hat{\boldsymbol{\tau}}^{23}\_{zy}(\mathbf{0}) = \mathbf{0}, \quad \hat{\boldsymbol{\upsilon}}^{32}(\mathbf{0}) = \hat{\boldsymbol{\upsilon}}^{34}(\mathbf{0}), \quad \hat{\boldsymbol{\tau}}^{32}\_{zy}(\mathbf{0}) + \hat{\boldsymbol{\tau}}^{34}\_{zy}(\mathbf{0}) = \mathbf{0} \tag{6}
$$

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 271

1 1 2 2 3 3

2 i 2 i

*s s*

 

 

> 

*h*

22 2

 

*h h*

1 e )(1 e )

3 3 1 1 2 2

*s ss*

 

*ss s*

. (11)

  (12)

*h*

(13)

(14)

and wavenumber *k* .

*k c*

in an isotropic medium *j* , which

 

The dispersion equation governing the characteristic SH waves in periodic ternary layered

det[ ( , , )] *<sup>d</sup>* **R 0** *k q* 

Further expansion of the determinant in Eq. (11) gives the closed-form dispersion relation of

i 2i 2 i 2i

*qh h*

3 3 1 1

2 2 i 2 i 2 i 2i

*qh hhh*

( ) ( ) e (1 e )(1 e )(1 e )

1 1 2 2 3 3

and <sup>i</sup> e 0 *qh* , Equation (12) can be simplified, by virtue of relations between

*qh h h h*

*ss s ss s*

 

> 

 

*hh h*

*hhh*

*ss s*

*qh hhh*

*qh hhh*

*ss s*

2 2

( ) ( ) e (1 e )(1 e )(1 e ) 0.

*s*

2 2 i 2 i 2 i 2 i

Dispersion equation (12) assures unconditionally numerical stability because Re[ i ] 0 *sj j*

is already guaranteed by the properly established phase relation and Re[i ] 0 *qh* can also be guaranteed by properly specifying *q* in the solving process, since *q* and *q* signifying wavenumbers in opposite direction should give the same propagation characteristics. Due

31 32 33 11 22 33

2 cos( ) cos( )cos( )cos( ) ( ) ( ) sin( )sin( )cos( )

 

 

 

<sup>3</sup> /i / / *ZSHj j sj j G G kG jj j*

 

as the characteristic impedance of SH wave in layer *j* , which is dependent on not only the

Therefore, the characteristic impedance of SH wave in an isotropic layer is not a constant and can be imaginary below the cutoff frequency of SH wave. It is very different from the

is a real-valued constant. But *ZSHj* equals to *ZTj* as 0 *k* . If 123 0 *ZZZ SH SH SH* , i.e.

regardless of *qh* . This case merely gives the cutoff frequency of SH wave mode *c s*

*kc kc kc* , the dispersion equation (13) is automatically satisfied

2 2 ). *<sup>s</sup>*

31 32 33 1 1 2 2 3 3

32 33 31 2 2 3 3 1 1

*s s*

*h h*

 

, but also the frequency

( ) ( ) sin( )sin( )cos( )

33 31 32 3 3 1 1

( ) ( ) sin( )sin( )cos(

media is obtained by vanishing of the determinant of system matrix

characteristic SH waves in periodic ternary layered media as follows

8 e e e (1 e )

*ss s*

2 2 i 2 i

( ) ( ) e (1 e )(

31 32 33

to <sup>i</sup> e 0 *sj j h*

Define

31 32 33

31 32 33

 

 

 

32 33 31

33 31 32

trigonometric functions and exponential functions, to

shear modulus *Gj* and mass density *<sup>j</sup>*

<sup>123</sup> ( )( )( ) 0 *sss* 

2 2

 

2 2

 

2 2

 

characteristic impedance of bulk shear wave *Z G Tj j j*

11 22 33

ii i 2i

2 e (1 e )(1 e )(1 e )

*hhh qh*

 

Substituting Eqs. (1) and (2) into Eq. (6), one obtains the scattering relations at interfaces 2 and 3

$$\begin{aligned} a\_3^{21} + d\_3^{21} - a\_3^{23} - d\_3^{23} &= 0, & \zeta\_3^{21} d\_3^{21} - \zeta\_3^{21} d\_3^{21} + \zeta\_3^{23} a\_3^{23} - \zeta\_3^{23} d\_3^{23} &= 0\\ a\_3^{32} + d\_3^{32} - a\_3^{34} - d\_3^{34} &= 0, & \zeta\_3^{32} a\_3^{32} - \zeta\_3^{32} d\_3^{32} + \zeta\_3^{34} a\_3^{34} - \zeta\_3^{34} d\_3^{34} &= 0 \end{aligned} \tag{7}$$

Third, we consider the spectral equations at the top and bottom surfaces. The Floquet-Bloch principle of periodic structures (Brillouin, 1953; Mead, 1996) requires that the displacement (stress) of bottom layer at the bottom surface 4 should relate to that of top layer at the top surface 1 by

$$
\hat{\boldsymbol{v}}^{43}(0) = \mathbf{e}^{\dagger \eta h} \,\hat{\boldsymbol{v}}^{12}(0), \quad \hat{\boldsymbol{\tau}}^{43}\_{zy}(0) = -\mathbf{e}^{\dagger \eta h} \,\hat{\boldsymbol{\tau}}^{12}\_{zy}(0) \tag{8}
$$

where <sup>3</sup> <sup>1</sup> *<sup>j</sup> <sup>j</sup> h h* is the thickness of the unit cell, *q* is the wavenumber of the characteristic waves in the periodic ternary layered media. The real part *Rq h* and the imaginary part *<sup>I</sup> q h* of dimensionless wavenumber *qh* denote the phase constant and the attenuation constant of the characteristic wave, respectively (Mead, 1996). Substitution of Eqs. (1) and (2) into Eq. (8) gives the scattering relations at surfaces 1 and 4

$$\mathbf{e}^{\mathrm{i}\neq\mathrm{h}}d\_{3}^{12} + \mathbf{e}^{\mathrm{i}\neq\mathrm{h}}d\_{3}^{12} - d\_{3}^{43} - d\_{3}^{43} = 0,\quad \mathbf{e}^{\mathrm{i}\neq\mathrm{h}}\boldsymbol{\zeta}^{12}\_{3}d\_{3}^{12} - \mathbf{e}^{\mathrm{i}\neq\mathrm{h}}\boldsymbol{\zeta}^{12}\_{3}d\_{3}^{12} + \boldsymbol{\zeta}^{43}\_{3}d\_{3}^{43} - \boldsymbol{\zeta}^{43}\_{3}d\_{3}^{43} = 0\tag{9}$$

Finally, introducing the phase relations of all layers as given in Eq. (5) to the scattering relations of all interfaces and surfaces as given in Eqs. (7) and (9), we obtain the system equations with all the departing wave amplitudes as basic unknown quantities

$$
\begin{bmatrix} P\_3^{21} & 1 & -1 & -P\_3^{23} & 0 & 0 \\ \zeta\_3^{21} P\_3^{21} & -\zeta\_3^{21} & -\zeta\_3^{23} & \zeta\_3^{23} P\_3^{23} & 0 & 0 \\ 0 & 0 & P\_3^{32} & 1 & -1 & -P\_3^{34} \\ 0 & 0 & \zeta\_3^{23} P\_3^{32} & -\zeta\_3^{32} & -\zeta\_3^{34} & \zeta\_3^{34} P\_3^{34} \\ \mathbf{e}^{1 \not\not\mathbf{f}} & \mathbf{e}^{1 \not\mathbf{f}} P\_3^{12} & 0 & 0 & -P\_3^{43} & -1 \\ -\zeta\_3^{12} \mathbf{e}^{1 \not\mathbf{f}} & \zeta\_3^{12} \mathbf{e}^{1 \not\mathbf{f}} P\_3^{12} & 0 & 0 & \zeta\_3^{43} P\_3^{43} & -\zeta\_3^{43} \\ \mathbf{e} & \zeta\_3^{12} \mathbf{e}^{1 \not\mathbf{f}} & \zeta\_3^{12} \mathbf{e}^{1 \not\mathbf{f}} P\_3^{12} & 0 & 0 & \zeta\_3^{43} P\_3^{43} & -\zeta\_3^{43} \\ \text{or} \\ \mathbf{R}\_d(\boldsymbol{\omega}, \boldsymbol{\alpha}, \boldsymbol{\eta}) \mathbf{d} & \mathbf{0} \end{bmatrix} \tag{10}$$

where 333 *JK KJ j* ( *JK j* 1 1,2,3 ) is the shear stress coefficient of layer *j* , **R***d* is the system matrix, and **d** is the global departing wave vector.

The dispersion equation governing the characteristic SH waves in periodic ternary layered media is obtained by vanishing of the determinant of system matrix

$$\det[\mathbf{R}\_d(k,\alpha,q)] = \mathbf{0} \,. \tag{11}$$

Further expansion of the determinant in Eq. (11) gives the closed-form dispersion relation of characteristic SH waves in periodic ternary layered media as follows

11 22 33 1 1 2 2 3 3 1 1 2 2 3 3 2 2 ii i 2i 31 32 33 i 2i 2 i 2i 31 32 33 2 2 i 2 i 2 i 2i 31 32 33 2 2 i 2 i 32 33 31 8 e e e (1 e ) 2 e (1 e )(1 e )(1 e ) ( ) ( ) e (1 e )(1 e )(1 e ) ( ) ( ) e (1 e )( *ss s ss s ss s s hhh qh qh hhh qh hhh qh h* 3 3 1 1 3 3 1 1 2 2 2 i 2 i 2 2 i 2 i 2 i 2 i 33 31 32 1 e )(1 e ) ( ) ( ) e (1 e )(1 e )(1 e ) 0. *s s s ss h h qh hhh* (12)

Dispersion equation (12) assures unconditionally numerical stability because Re[ i ] 0 *sj j h* is already guaranteed by the properly established phase relation and Re[i ] 0 *qh* can also be guaranteed by properly specifying *q* in the solving process, since *q* and *q* signifying wavenumbers in opposite direction should give the same propagation characteristics. Due to <sup>i</sup> e 0 *sj j h* and <sup>i</sup> e 0 *qh* , Equation (12) can be simplified, by virtue of relations between trigonometric functions and exponential functions, to

$$\begin{aligned} &2\zeta\_{31}\zeta\_{32}\zeta\_{33}\left[\cos(qh) - \cos(\chi\_{s1}h\_1)\cos(\chi\_{s2}h\_2)\cos(\chi\_{s3}h\_3)\right] = \\ &-\left[\left(\zeta\_{31}\right)^2 + \left(\zeta\_{32}\right)^2\right]\zeta\_{33}\sin(\chi\_{s1}h\_1)\sin(\chi\_{s2}h\_2)\cos(\chi\_{s3}h\_3) \\ &-\left[\left(\zeta\_{32}\right)^2 + \left(\zeta\_{33}\right)^2\right]\zeta\_{31}\sin(\chi\_{s2}h\_2)\sin(\chi\_{s3}h\_3)\cos(\chi\_{s1}h\_1) \\ &-\left[\left(\zeta\_{33}\right)^2 + \left(\zeta\_{31}\right)^2\right]\zeta\_{32}\sin(\chi\_{s3}h\_3)\sin(\chi\_{s1}h\_1)\cos(\chi\_{s2}h\_2) .\end{aligned} \tag{13}$$

Define

270 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

 

(8) gives the scattering relations at surfaces 1 and 4

0 0

 

(, ,) *<sup>d</sup> k q* 

**R d0**

*j*

or

where 333 *JK KJ*

*qh qh qh qh*

 

expressed as

surface 1 by

where <sup>3</sup>

and 3

Second, we consider the spectral equations at the interfaces between adjacent layers. The compatibility of displacements and equilibrium of stresses at the interfaces 2 and 3 are

21 23 21 23 32 34 32 34 ˆ ˆ (0) (0), (0) (0) 0, (0) (0), (0) (0) 0 ˆ ˆ ˆ ˆ ˆ ˆ *zy zy zy zy v v*

Substituting Eqs. (1) and (2) into Eq. (6), one obtains the scattering relations at interfaces 2

21 21 23 23 21 21 21 21 23 23 23 23 3 3 3 3 33 33 33 33 32 32 34 34 32 32 32 32 34 34 34 34 3 3 3 3 33 33 33 33

 

Third, we consider the spectral equations at the top and bottom surfaces. The Floquet-Bloch principle of periodic structures (Brillouin, 1953; Mead, 1996) requires that the displacement (stress) of bottom layer at the bottom surface 4 should relate to that of top layer at the top

> <sup>43</sup> i i 12 43 <sup>12</sup> ˆ ˆ (0) e (0), (0) e (0) ˆ ˆ *qh qh zy zy v v*

waves in the periodic ternary layered media. The real part *Rq h* and the imaginary part *<sup>I</sup> q h* of dimensionless wavenumber *qh* denote the phase constant and the attenuation constant of the characteristic wave, respectively (Mead, 1996). Substitution of Eqs. (1) and (2) into Eq.

> i i 12 12 43 43 i i 12 12 12 12 43 43 43 43 3 333 33 33 33 33 e e 0, e e <sup>0</sup> *qh qh qh qh a dad a d a d*

21 23 12 3 3 3 21 21 21 23 23 23 21 33 3 3 33 3

*P P d P P d*

1 1 00

i i 12 43 34

 

e e 00 1

12 12 12 i i 43 43 43 43 3 33 33 3 3

3 3 3

*P P d P P d*

equations with all the departing wave amplitudes as basic unknown quantities

0 0 11

 

e e 00

the system matrix, and **d** is the global departing wave vector.

Finally, introducing the phase relations of all layers as given in Eq. (5) to the scattering relations of all interfaces and surfaces as given in Eqs. (7) and (9), we obtain the system

<sup>1</sup> *<sup>j</sup> <sup>j</sup> h h* is the thickness of the unit cell, *q* is the wavenumber of the characteristic

*adad a d a d adad a d a d*

*v v*

0, 0 0, 0

> 

32 34 23 3 3 3 32 32 32 34 34 34 32 33 3 3 33 3

*P P d P P d*

 

( *JK j* 1 1,2,3 ) is the shear stress coefficient of layer *j* , **R***d* is

0 0

 

 

(8)

(6)

(7)

(9)

(10)

$$Z\_{\rm SHj} = \mathcal{L}\_{3j} \;/\; \mathbf{i} \; \alpha = \mathcal{V}\_{s\underline{j}} \mathbf{G}\_{\underline{j}} \;/\; \alpha = \sqrt{\rho\_{\underline{j}} \mathbf{G}\_{\underline{j}} - \mathbf{k}^2 \mathbf{G}\_{\underline{j}}^2 / \; \alpha^2} \tag{14}$$

as the characteristic impedance of SH wave in layer *j* , which is dependent on not only the shear modulus *Gj* and mass density *<sup>j</sup>* , but also the frequency and wavenumber *k* . Therefore, the characteristic impedance of SH wave in an isotropic layer is not a constant and can be imaginary below the cutoff frequency of SH wave. It is very different from the characteristic impedance of bulk shear wave *Z G Tj j j* in an isotropic medium *j* , which is a real-valued constant. But *ZSHj* equals to *ZTj* as 0 *k* . If 123 0 *ZZZ SH SH SH* , i.e. <sup>123</sup> ( )( )( ) 0 *sss kc kc kc* , the dispersion equation (13) is automatically satisfied regardless of *qh* . This case merely gives the cutoff frequency of SH wave mode *c s k c*

within one of the constituent layers, and we shall not discuss it any further. Otherwise for <sup>123</sup> 0 *ZZZ SH SH SH* , the dispersion relation is further simplified to

$$\begin{aligned} \cos(\eta h) &= \cos(\chi\_{s1}h\_1)\cos(\chi\_{s2}h\_2)\cos(\chi\_{s3}h\_3) - \frac{1}{2}(F\_{1/2} + F\_{2/1})\sin(\chi\_{s1}h\_1)\sin(\chi\_{s2}h\_2)\cos(\chi\_{s3}h\_3) \\ &- \frac{1}{2}(F\_{2/3} + F\_{3/2})\sin(\chi\_{s2}h\_2)\sin(\chi\_{s3}h\_3)\cos(\chi\_{s1}h\_1) - \frac{1}{2}(F\_{3/1} + F\_{1/3})\sin(\chi\_{s3}h\_3)\sin(\chi\_{s1}h\_1)\cos(\chi\_{s2}h\_2). \end{aligned} \tag{15}$$

where

$$F\_{\mathbf{j'}\mathbf{j''}} = \frac{\mathcal{L}\_{\mathbf{3j'}}}{\mathcal{L}\_{\mathbf{3j''}}} = \frac{\mathbf{i}\,\gamma\_{\mathbf{s}\mathbf{j'}}\mathbf{G}\_{\mathbf{j''}}}{\mathbf{i}\,\gamma\_{\mathbf{s}\mathbf{j''}}\mathbf{G}\_{\mathbf{j''}}} = \frac{\gamma\_{\mathbf{s}\mathbf{j'}}\mathbf{G}\_{\mathbf{j'}}\,/\,\mathrm{co}}{\gamma\_{\mathbf{s}\mathbf{j'}}\mathbf{G}\_{\mathbf{j''}}\,/\,\mathrm{co}} = \frac{\mathbf{Z}\_{\mathrm{SH}\mathbf{j'}}}{\mathbf{Z}\_{\mathrm{SH}\mathbf{j'}}} \quad \left(\mathbf{j'}\,\mathrm{j''} = \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{j'} \neq \mathbf{j''}\right) \tag{16}$$

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 273

*<sup>c</sup>*max , and the characteristic time of the unit cell decides the

 

*h hh* and *ij j hh h* are the wavenumber

*<sup>c</sup>*max the characteristic impedances and the wavenumbers

2

(21)

  *b*

as *Z Z SHj SHj* . Therefore, the dispersion

 

*b*

) when / 1 *<sup>j</sup> F* , / 0 1 *<sup>j</sup>*

*j j*

 

*<sup>c</sup>* max .

(19)

the fundamental dispersion curve of the equivalent SH wave in the unit cell due to the zone folding effect (Brillouin, 1953) with the characteristic time of the unit cell being the essential parameter. In other words, the contrast of characteristic impedances determines whether the

/ /

*sj j sj j sj j j j j j sj j sj j sj j*

band gaps exist or not above

If *Z Z SHj SHj* , then the dispersion relation (15) is reduced to

 

 

the dispersion relation should be the general form of Eq. (15).

 

 ) when / <sup>1</sup> *<sup>j</sup> <sup>F</sup>* , *j jj* min( , ) (0,1) / / 

as *Z Z SHj SHj* , 0 1 *j j*

equations (15) and (19) can be rewritten uniformly as

Cao, 2000; Wang et al., 2004). ( ) / *si sj j sj j i*

In any circumstances, as

 and *<sup>j</sup>* 

Thus, we have

*F F*

where *<sup>j</sup>*/ , / *<sup>j</sup>* 

( / 1 *<sup>j</sup>* 

where 0 *j j* 

dispersion spectra of the periodic layered media as no bandgap exists above

2. , *Z ZZ Z SHj SHj SHj SHj* ( *Z or Z SHj SHj* , *jj j* , , 1 or 2 or 3 , *jj j* )

/ /

*h h FF h h*

*sj j si i j i i j sj j si i*

 

<sup>1</sup> cos( )cos( ) ( )sin( )sin( ), <sup>2</sup>

<sup>1</sup> cos( ) cos( )cos( ) ( )sin( )sin( ) <sup>2</sup>

which is the dispersion relation for periodic binary layered media already obtained (Shen &

along the *z* -direction and the thickness of the new equivalent layer *i* composed of the two constituent layers with identical characteristic impedance of SH wave. If *Z Z SHj SHj* , then

along the *z* -direction of SH waves in the constituent layers will be positive real number.

// / /

*j j j j jj j j*

*j jj j j j*

are real numbers, / 0 1 *<sup>j</sup>*

 

 

/ /

*F F*

/ / 2 1 1 <sup>1</sup> <sup>1</sup> 11 2 , 1 1 <sup>1</sup>

> ( / 1 *<sup>j</sup>*

 (20)

*j j j b*

. Similarly, we have

<sup>1</sup> 1 2, <sup>1</sup>

*j j b*

*qh h h h F F h h h*

signifies the contrast of characteristic impedances of SH waves in layer *j* and layer *j* .

#### **2.2. Formation mechanisms of SH-wave bands in the Bragg Cell**

Based on Eq. (15), in which 123 0 *ZZZ SH SH SH* is implied, in what follows we will discuss the formation mechanisms of frequency bands of SH waves in periodic ternary layered media, according to the following two cases for the characteristic impedances of SH waves in the three constituent layers of the unit cell.

$$1.\qquad Z\_{SH1} = Z\_{SH2} = Z\_{SH3}$$

Owing to / / 1 *jj jj F F* ( *j j* , 1,2,3 , *j j* ), the dispersion relation (15) is reduced to

$$\begin{aligned} \cos(qh) &= \cos(\gamma\_{s1}h\_1 + \gamma\_{s2}h\_2 + \gamma\_{s3}h\_3) = \cos(\gamma\_{sc}h) \\ &= \cos(\frac{\alpha}{c\_{SH1}}h\_1 + \frac{\alpha}{c\_{SH2}}h\_2 + \frac{\alpha}{c\_{SH3}}h\_3) = \cos[\alpha(T\_{SH1} + T\_{SH2} + T\_{SH3})] = \cos(\alpha T\_{SH}), \end{aligned} \tag{17}$$

where <sup>3</sup> <sup>1</sup>( )/ *se j sj j h h* is the equivalent wavenumber of SH wave in the unit cell, / *SHj j SHj T hc* is the parameter reflecting the characteristic time as SH wave traverses the thickness of constituent *j*, but may be imaginary number below the cutoff frequency *c* , 3 *SH* <sup>1</sup> *SHj <sup>j</sup> T T* is the parameter reflecting the characteristic time as SH wave traverses the thickness of the unit cell. Equation (17) has the solution

$$
\mu \eta h = \pm \gamma\_{\text{se}} h + 2 \frac{m}{n} \pi = \pm \alpha \Gamma\_{SH} + 2 \frac{m}{n} \pi \,, \tag{18}
$$

where *m* and *n* are arbitrary integers corresponding to positive and negative signs, respectively . Equation (18) indicates that when all the three constituent layers have the same characteristic impedance of SH wave, there is no bandgap above the maximum cutoff frequency *<sup>c</sup>*max max( , , ) *ccc* <sup>123</sup> . The dispersion spectra are completely determined by the fundamental dispersion curve of the equivalent SH wave in the unit cell due to the zone folding effect (Brillouin, 1953) with the characteristic time of the unit cell being the essential parameter. In other words, the contrast of characteristic impedances determines whether the band gaps exist or not above *<sup>c</sup>*max , and the characteristic time of the unit cell decides the dispersion spectra of the periodic layered media as no bandgap exists above *<sup>c</sup>* max .

$$\text{2.}\quad Z\_{\text{SH}\downarrow} \neq Z\_{\text{SH}\uparrow}Z\_{\text{SH}\downarrow} \neq Z\_{\text{SH}\uparrow} \text{ ( $Z\_{\text{SH}\uparrow} = \text{or } \neq Z\_{\text{SH}\uparrow}$ ,  $\not\llcorner\uparrow\downarrow$  '> \text{'} = 1 \text{ or } \text{2 or } \text{3 } \text{'} \text{ ( $\not\llfil\uparrow\neq \text{''}$ )}$$

If *Z Z SHj SHj* , then the dispersion relation (15) is reduced to

272 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

<sup>123</sup> 0 *ZZZ SH SH SH* , the dispersion relation is further simplified to

 

2 2

11 22 33

*qh h h h h*

cos( ) cos( ) cos( )

thickness of the unit cell. Equation (17) has the solution

12 3

*SH SH SH*

*cc c*

*ss s se*

3

/

three constituent layers of the unit cell.

<sup>1</sup>( )/ *se j sj j*

 

1. *ZZZ SH SH SH* <sup>123</sup>

where <sup>3</sup>

3

frequency

*j j*

where

within one of the constituent layers, and we shall not discuss it any further. Otherwise for

1 1 2 2 3 3 1/2 2/1 1 1 2 2 3 3

<sup>3</sup>

(16)

, 1,2,3, i /

*ss s sss*

 

(15)

*SH SH SH SH*

 

(18)

(17)

*c* ,

*ss s ss s*

2/3 3/2 2 2 3 3 1 1 3/1 1/3 3 3 1 1 2 2

1 1 ( )sin( )sin( )cos( ) ( )sin( )sin( )cos( ),

*FF h h h FF h h h*

 

Based on Eq. (15), in which 123 0 *ZZZ SH SH SH* is implied, in what follows we will discuss the formation mechanisms of frequency bands of SH waves in periodic ternary layered media, according to the following two cases for the characteristic impedances of SH waves in the

Owing to / / 1 *jj jj F F* ( *j j* , 1,2,3 , *j j* ), the dispersion relation (15) is reduced to

 

/ *SHj j SHj T hc* is the parameter reflecting the characteristic time as SH wave traverses the

*SH* <sup>1</sup> *SHj <sup>j</sup> T T* is the parameter reflecting the characteristic time as SH wave traverses the

*m m qh h <sup>T</sup>*

where *m* and *n* are arbitrary integers corresponding to positive and negative signs, respectively . Equation (18) indicates that when all the three constituent layers have the same characteristic impedance of SH wave, there is no bandgap above the maximum cutoff

2 2, *se SH*

*n n*

*<sup>c</sup>*max max( , , ) *ccc* <sup>123</sup> . The dispersion spectra are completely determined by

 

thickness of constituent *j*, but may be imaginary number below the cutoff frequency

1 2 3 123

cos( ) cos[ ( )] cos( ),

*h h h TTT T*

*h h* is the equivalent wavenumber of SH wave in the unit cell,

<sup>1</sup> cos( ) cos( )cos( )cos( ) ( )sin( )sin( )cos( ) <sup>2</sup>

i /

**2.2. Formation mechanisms of SH-wave bands in the Bragg Cell** 

*j sj j sj j SHj*

 

 

*GG Z <sup>F</sup> jj j j GG Z*

signifies the contrast of characteristic impedances of SH waves in layer *j* and layer *j* .

*j sj j sj j SHj*

*qh h h h F F h h h*

$$\begin{split} \cos(qh) &= \cos(\chi\_{s\uparrow}h\_{\uparrow})\cos(\chi\_{s\uparrow}h\_{\uparrow}+\chi\_{s\uparrow}h\_{\uparrow}\prime) - \frac{1}{2}(F\_{\uparrow\mid\uparrow}+F\_{\uparrow\mid\uparrow})\sin(\chi\_{s\uparrow}h\_{\uparrow})\sin(\chi\_{s\uparrow}h\_{\uparrow}+\chi\_{s\uparrow}h\_{\uparrow}\prime) \\ &= \cos(\chi\_{s\uparrow}h\_{\uparrow})\cos(\chi\_{s\downarrow}h\_{\uparrow}) - \frac{1}{2}(F\_{\uparrow\mid\uparrow}+F\_{\uparrow\mid\uparrow})\sin(\chi\_{s\uparrow}h\_{\uparrow})\sin(\chi\_{s\uparrow}h\_{\uparrow}) . \end{split} \tag{19}$$

which is the dispersion relation for periodic binary layered media already obtained (Shen & Cao, 2000; Wang et al., 2004). ( ) / *si sj j sj j i h hh* and *ij j hh h* are the wavenumber along the *z* -direction and the thickness of the new equivalent layer *i* composed of the two constituent layers with identical characteristic impedance of SH wave. If *Z Z SHj SHj* , then the dispersion relation should be the general form of Eq. (15).

In any circumstances, as *<sup>c</sup>*max the characteristic impedances and the wavenumbers along the *z* -direction of SH waves in the constituent layers will be positive real number. Thus, we have

$$F\_{j/\Gamma} + F\_{\Gamma/j} = 1 - \varepsilon\_{j/\Gamma} + \frac{1}{1 - \varepsilon\_{j/\Gamma}} = \frac{1}{1 - \varepsilon\_{\Gamma/j}} + 1 - \varepsilon\_{\Gamma/j} = 1 - \varepsilon\_{j\Gamma} + \frac{1}{1 - \varepsilon\_{j\Gamma}} = 2 + \sum\_{b=2}^{+\sigma} \varepsilon\_{j\Gamma}^b \quad \left(\Gamma = j', j''\right) \tag{20}$$

where *<sup>j</sup>*/ , / *<sup>j</sup>* and *<sup>j</sup>* are real numbers, / 0 1 *<sup>j</sup>* ( / 1 *<sup>j</sup>* ) when / 1 *<sup>j</sup> F* , / 0 1 *<sup>j</sup>* ( / 1 *<sup>j</sup>* ) when / <sup>1</sup> *<sup>j</sup> <sup>F</sup>* , *j jj* min( , ) (0,1) / / . Similarly, we have

$$F\_{"/\prime\prime"} + F\_{"/\prime\prime"} = 1 - \varepsilon\_{\prime\prime\prime"} + \frac{1}{1 - \varepsilon\_{\prime\prime\prime"}} = 2 + \sum\_{b=2}^{+\infty} \varepsilon\_{\prime\prime\prime"}^{b} \tag{21}$$

where 0 *j j* as *Z Z SHj SHj* , 0 1 *j j* as *Z Z SHj SHj* . Therefore, the dispersion equations (15) and (19) can be rewritten uniformly as

2 2 2 2 <sup>1</sup> cos( ) cos( ) sin( )sin( )cos( ) <sup>2</sup> 1 1 sin( )sin( )cos( ) sin( )sin( )cos( ) 2 2 cos( ) sin( 2(1 ) *b se jj sj j sj j sj j b b b jj sj j sj j sj j j j sj j sj j sj j b b jj se jj qh h hh h hh h h h h h* 2 2 2 )sin( )cos( ) sin( )sin( )cos( ) sin( )sin( )cos( ) 2(1 ) 2(1 ) cos( ) sin( )sin( )cos( ) 2(1 ) *sj j sj j sj j jj j j sj j sj j sj j sj j sj j sj j jj j j jj SH SHj SHj SHj jj j hh h hh h hhh T TT T* 2 2 sin( )sin( )cos( ) sin( )sin( )cos( ), 2(1 ) 2(1 ) *j j j SHj SHj SHj SHj SHj SHj jj j j TT T TT T* (22)

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 275

*h* denote the phase changes as SH wave passes through

In physics, at any interface *J* of the unit cell there are one incident wave *wiJ* and one reflected wave *wrj* arising from the next interface except that there is no reflection at the interface where the two constituent layers with identical characteristic impedances

layer *j* , *j* and *j* , respectively. Thus, the former formula in Eq. (23) corresponds to the constructive interference condition of the incident wave and reflected wave at two interfaces, and the latter formula in Eq. (23) corresponds to the destructive interference condition at three interfaces. Equation (24) corresponds to destructive interference condition of incident wave and reflected wave at two interfaces and constructive interference condition at one interface. Therefore, it is concluded that the frequency bands are formed physically as a result of interference phenomenon as waves transmit and reflect in the constituent layers of a periodic ternary layered media. The specified combination of exact constructive and destructive interferences of the incident and reflected waves at some interfaces makes the equivalent SH wave travel through the unit cell without any change of its dispersion characteristic or be completely prohibited to travel. The specified combination of near constructive and destructive interferences of the incident and reflected waves at some interfaces makes the equivalent SH wave be capable of going through the unit cell with a change of its dispersion characteristic or be attenuated. The exact constructive and destructive interferences specified by Eq. (23) and Eq. (24) are only possible for special periodic ternary layered media with the characteristic times of constituent layers satisfying

: : or : : (2 1) : (2 1) : (2 1) *SHj SHj j j SHj SHj SHj j <sup>j</sup> <sup>j</sup> T T gg T T T g g g* (25)

However, the near constructive and destructive interferences specified by Eq. (23) and Eq.

In summary, the occurrence of some specified combination of exact or near constructive and destructive interference phenomena in the unit cell makes the equivalent SH wave travel through the unit cell and gives birth to the pass-bands, whereas the occurrence of other specified combination of exact or near destructive interference makes the equivalent SH wave unable to pass through the unit cell and brings about the stop-bands. Although the above discussion on the formation mechanisms of SH-wave bands is based on the periodic ternary layered structure, it is actually extendable to SH wave in general periodic layered media.

The discussion of formation mechanisms of SH wave bands in the layered Bragg Cell indicates that the contrasts of characteristic impedances of the constituent layers, the characteristic time of the unit cell and the characteristic times of the constituent layers are three kinds of essential parameters, which influence the band properties. First, the contrasts of characteristic impedances decide whether the stop-bands other than that due to SH wave

**2.3. Design rules to the Bragg Cell concerning with SH wave bands** 

: : : (2 1) : (2 1) *SHj SHj SHj j j <sup>j</sup> TT T g g g* (26)

connected. *sj j* 

 *h* , *sj j* 

 *h* and *sj j* 

(24) can occur in general periodic ternary layered media.

which indicates that the band structures of the periodic ternary layered media are not only determined by the fundamental dispersion curve of the equivalent SH wave according to zone folding effect, but also influenced by three disturbance terms with disturbing functions sin( )sin( )cos( ) *sj j sj j sj j hhh* (or sin( )sin( )cos( ) *SHj SHj SHj TT T* , *jj j* , , 1,2,3 , *jj j* ) and disturbing amplitudes <sup>2</sup> /[2(1 )] *j j j j* . The value of the right-hand side of Eq. (22) determines the demarcation of frequency bands: 1 gives the dividing lines of pass-bands and stop-bands; 0 gives the central frequencies of pass-bands; those between 1 and 1 give the pass-bands; and all other values give the stop-bands. The characteristic time of the unit cell, the characteristic times of constituent layers and the contrasts of characteristic impedances of SH waves in the constituent layers are the essential parameters for the band structure formation, which determine the shape of the dispersion curves of the equivalent SH wave (the pre-disturbed baselines), the shapes of the disturbing functions, and the amplitudes of the disturbance terms, respectively. When the disturbing functions satisfy sin( )sin( )cos( ) 0 *sj j sj j sj j hhh* ( sin( )sin( )cos( ) 1 *sj j sj j sj j hhh* ), the band structures coincide exactly with (deviate most from) the fundamental and derivative dispersion curves of the equivalent SH wave in the unit cell. The corresponding points on the dispersion curves are called as the coincident (separating) points. The frequency equation can be simplified to ( *jj j* , , 1,2,3 , *jj j* )

$$\begin{aligned} \gamma\_{s\uparrow}h\_{\uparrow} &= \mathcal{g}\_{\uparrow}\pi\_{\star} \; \gamma\_{s\uparrow}h\_{\uparrow} = \mathcal{g}\_{\uparrow}\pi \; \left(\mathcal{g}\_{\uparrow} \text{ and } \mathcal{g}\_{\uparrow} \text{ are integers}\right) \text{or} \\ \gamma\_{s\uparrow}h\_{\uparrow} &= (\mathcal{2}\mathcal{g}\_{\uparrow} + 1)\pi \; \left(\mathcal{2}, \; \gamma\_{s\uparrow}h\_{\uparrow} = (\mathcal{2}\mathcal{g}\_{\uparrow} + 1)\pi \; \mid \; \mathcal{2}, \; \gamma\_{s\uparrow}h\_{\uparrow} = (\mathcal{2}\mathcal{g}\_{\uparrow} + 1)\pi \; \mid \; \mathcal{2} \end{aligned} \tag{23}$$

$$\mathbf{g}\_{s\circ}\mathbf{h}\_{\circ} = \mathbf{g}\_{\circ}\boldsymbol{\pi}\_{\circ}\ \boldsymbol{\gamma}\_{s\circ}\mathbf{h}\_{\circ} = (\mathbf{2}\,\mathbf{g}\_{\circ^\*} + \mathbf{1})\boldsymbol{\pi}\ /\ \mathbf{2},\ \boldsymbol{\gamma}\_{s\circ}\mathbf{h}\_{\circ^\*} = (\mathbf{2}\,\mathbf{g}\_{\circ^\*} + \mathbf{1})\boldsymbol{\pi}\ /\ \mathbf{2}\,,\tag{24}$$

for coincident and separating points, respectively.

In physics, at any interface *J* of the unit cell there are one incident wave *wiJ* and one reflected wave *wrj* arising from the next interface except that there is no reflection at the interface where the two constituent layers with identical characteristic impedances connected. *sj j h* , *sj j h* and *sj j h* denote the phase changes as SH wave passes through layer *j* , *j* and *j* , respectively. Thus, the former formula in Eq. (23) corresponds to the constructive interference condition of the incident wave and reflected wave at two interfaces, and the latter formula in Eq. (23) corresponds to the destructive interference condition at three interfaces. Equation (24) corresponds to destructive interference condition of incident wave and reflected wave at two interfaces and constructive interference condition at one interface. Therefore, it is concluded that the frequency bands are formed physically as a result of interference phenomenon as waves transmit and reflect in the constituent layers of a periodic ternary layered media. The specified combination of exact constructive and destructive interferences of the incident and reflected waves at some interfaces makes the equivalent SH wave travel through the unit cell without any change of its dispersion characteristic or be completely prohibited to travel. The specified combination of near constructive and destructive interferences of the incident and reflected waves at some interfaces makes the equivalent SH wave be capable of going through the unit cell with a change of its dispersion characteristic or be attenuated. The exact constructive and destructive interferences specified by Eq. (23) and Eq. (24) are only possible for special periodic ternary layered media with the characteristic times of constituent layers satisfying

274 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

 

*se jj sj j sj j sj j*

 

1 1 sin( )sin( )cos( ) sin( )sin( )cos( ) 2 2

)sin( )cos( )

sin( )sin( )cos( ) sin( )sin( )cos( ) 2(1 ) 2(1 )

*sj j sj j sj j*

*hh h*

 

*jj sj j sj j sj j j j sj j sj j sj j*

*hh h h h h*

*sj j sj j sj j sj j sj j sj j*

 

> 

*SHj SHj SHj SHj SHj SHj*

 

 

*hhh* ), the band structures

 

(23)

, (24)

. The value of the right-hand side of Eq. (22)

*TT T TT T* 

> 

*hh h hhh*

 

 

 

sin( )sin( )cos( ) sin( )sin( )cos( ), 2(1 ) 2(1 )

which indicates that the band structures of the periodic ternary layered media are not only determined by the fundamental dispersion curve of the equivalent SH wave according to zone folding effect, but also influenced by three disturbance terms with disturbing functions

determines the demarcation of frequency bands: 1 gives the dividing lines of pass-bands and stop-bands; 0 gives the central frequencies of pass-bands; those between 1 and 1 give the pass-bands; and all other values give the stop-bands. The characteristic time of the unit cell, the characteristic times of constituent layers and the contrasts of characteristic impedances of SH waves in the constituent layers are the essential parameters for the band structure formation, which determine the shape of the dispersion curves of the equivalent SH wave (the pre-disturbed baselines), the shapes of the disturbing functions, and the amplitudes of the disturbance terms, respectively. When the disturbing functions satisfy

 

> 

> >

 

> 

*TT T* , *jj j* , , 1,2,3 , *jj j* )

 

> 

> >

(22)

2

 

*b*

 

> 

<sup>1</sup> cos( ) cos( ) sin( )sin( )cos( ) <sup>2</sup>

*qh h hh h*

*b b*

*b*

2 2

 

2 2

*jj j j*

cos( ) sin( )sin( )cos( ) 2(1 )

2 2

*j j j*

*T TT T*

*SH SHj SHj SHj*

*jj j j*

 

 

*hhh* (or sin( )sin( )cos( ) *SHj SHj SHj*

 

*hhh* ( sin( )sin( )cos( ) 1 *sj j sj j sj j*

*sj j j sj j j j j*

 

 

for coincident and separating points, respectively.

*hg hg g g*

 

coincide exactly with (deviate most from) the fundamental and derivative dispersion curves of the equivalent SH wave in the unit cell. The corresponding points on the dispersion curves are called as the coincident (separating) points. The frequency equation can be

, and are integers or

, (2 1) / 2, (2 1) / 2 *sj j j sj j j sj j j*

 *hg h g h g* 

*sj j j sj j j sj j j*

*hg h g h g*

(2 1) / 2, (2 1) / 2, (2 1) / 2

 

 

*jj j j*

*b b*

2

*jj*

 

*jj*

2

*jj*

*jj*

 

 

> 

and disturbing amplitudes <sup>2</sup> /[2(1 )] *j j j j*

cos( ) sin( 2(1 )

*se*

*j*

sin( )sin( )cos( ) *sj j sj j sj j*

sin( )sin( )cos( ) 0 *sj j sj j sj j*

simplified to ( *jj j* , , 1,2,3 , *jj j* )

*h*

$$T\_{\rm SH} \colon T\_{\rm SH} = \mathcal{g}\_{\sf f} \colon \mathcal{g}\_{\sf f} \quad \text{or} \quad T\_{\rm SH} \colon T\_{\rm SH} \colon T\_{\rm SH} = (\mathcal{2g}\_{\sf f} + 1) \colon (\mathcal{2g}\_{\sf f} + 1) \colon (\mathcal{2g}\_{\sf f} + 1) \tag{25}$$

$$T\_{\rm SHf} : T\_{\rm SHf^\*} : T\_{\rm SH^\*} = \mathbb{g}\_f : (\mathbb{2}\,\mathbb{g}\_{f^\*} + 1) : (\mathbb{2}\,\mathbb{g}\_{f^\*} + 1) \tag{26}$$

However, the near constructive and destructive interferences specified by Eq. (23) and Eq. (24) can occur in general periodic ternary layered media.

In summary, the occurrence of some specified combination of exact or near constructive and destructive interference phenomena in the unit cell makes the equivalent SH wave travel through the unit cell and gives birth to the pass-bands, whereas the occurrence of other specified combination of exact or near destructive interference makes the equivalent SH wave unable to pass through the unit cell and brings about the stop-bands. Although the above discussion on the formation mechanisms of SH-wave bands is based on the periodic ternary layered structure, it is actually extendable to SH wave in general periodic layered media.

#### **2.3. Design rules to the Bragg Cell concerning with SH wave bands**

The discussion of formation mechanisms of SH wave bands in the layered Bragg Cell indicates that the contrasts of characteristic impedances of the constituent layers, the characteristic time of the unit cell and the characteristic times of the constituent layers are three kinds of essential parameters, which influence the band properties. First, the contrasts of characteristic impedances decide whether the stop-bands other than that due to SH wave

cutoff property exist or not. When the characteristic impedances of all the constituent layers are identical, any SH waves above the maximum cutoff frequency can propagate in the periodic layer without attenuation and no stop-bands other than that due to the SH wave cutoff property exist. In other cases, stop-bands exist above the maximum cutoff frequency, and the contrasts of characteristic impedances decide the widths of the frequency bands. The characteristic time of the unit cell decides the slopes of the dispersion curves of equivalent SH waves, thus it definitely specifies the number of pass-bands/stop-bands in a given frequency range. The characteristic times of the constituent layers mainly decide the mid-frequencies of the frequency bands. It should be pointed out that the mass densities and shear moduli of constituent layers affect all the three kinds of essential parameters, while the thicknesses of the constituent layers only influence the characteristic times of the unit cell and of the constituent layers. These rules can be used for the design of layered Bragg Cells according to the SH-wave bands requirements.

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 277

## **2.4. Numerical examples**

In this section, the above proposed MRRM for dispersion characteristic analysis and the mechanisms for band structure formation of SH waves in periodic ternary layered media are validated by considering a periodic ternary layered structure with the unit cell consisting of one Pb layer in the middle and two epoxy layers at the up and down sides. The thickness of the Pb layer is 10mm and that of the epoxy layers is 5mm. The material parameters of Pb and epoxy including the Young's modulus, shear modulus and mass density are 40.8187 *Pb E* GPa, 4.35005 *epoxy E* GPa, 14.9 *GPb* GPa, 1.59 *Gepoxy* GPa, 11600 *Pb* kg/m3 and 1180 *epoxy* kg/m3. The band structures of SH waves in this periodic ternary layered medium are calculated by the formulation presented in Section 2.1. For the convenience of presentation, the results are represented by the dimensionless wavenumbers *kh* / and *qh* / with 0.02 *h* m being the total thickness of the unit cell, and the engineering frequency *f* / 2.

We first consider the property of SH-wave band structures in the exemplified periodic ternary layered medium. Figure 3 gives the band structures of SH waves below 140 kHz represented as various forms of graphs. Figure 3(a) depicts the phase constant surfaces in the pass-bands as the dimensionless wavenumbers *kh* / and *qh* / are in the range of [ 2,2] . Figures 3(b) to 3(d) describe both the phase constant spectra in pass-bands (i.e. the relation between *f* and / *Rq h* ) and the attenuation constant spectra in stop-bands (i.e. the relation between *f* and / *<sup>I</sup> q h* ) as the dimensionless wavenumber *kh* / is 1.0 , 0.5 and 0.0 , respectively. Figure 3(e) plots the relation between *f* and *k h* / when *qh* / is specified as 1 0.0 2 *I* , 2 0.5 2 *I* and 3 1.0 2 *I* with 1*I* , 2*I* and 3*I* being arbitrary integers. To validate our obtained results, in Fig. (3d) the phase constant spectra as / 0 *kh* are compared to the corresponding results calculated by Wang et al. (2004), and in Fig. (3e) the spectra of *f* versus *k h* / as 1 *qh I* / 0 2 and 3 *qh I* / 1 2 are compared to their counterparts calculated by Wang et al. (2004).

according to the SH-wave bands requirements.

**2.4. Numerical examples** 

kg/m3 and 1180 *epoxy* 

and *qh* /

engineering frequency *f*

relation between *f* and / *Rq h*

relation between *f* and / *<sup>I</sup> q h*

spectra of *f* versus *k h* /

 / 2.

the pass-bands as the dimensionless wavenumbers *kh* /

counterparts calculated by Wang et al. (2004).

0.0 , respectively. Figure 3(e) plots the relation between *f* and *k h* /

 as 1 *qh I* / 

*kh* / 

cutoff property exist or not. When the characteristic impedances of all the constituent layers are identical, any SH waves above the maximum cutoff frequency can propagate in the periodic layer without attenuation and no stop-bands other than that due to the SH wave cutoff property exist. In other cases, stop-bands exist above the maximum cutoff frequency, and the contrasts of characteristic impedances decide the widths of the frequency bands. The characteristic time of the unit cell decides the slopes of the dispersion curves of equivalent SH waves, thus it definitely specifies the number of pass-bands/stop-bands in a given frequency range. The characteristic times of the constituent layers mainly decide the mid-frequencies of the frequency bands. It should be pointed out that the mass densities and shear moduli of constituent layers affect all the three kinds of essential parameters, while the thicknesses of the constituent layers only influence the characteristic times of the unit cell and of the constituent layers. These rules can be used for the design of layered Bragg Cells

In this section, the above proposed MRRM for dispersion characteristic analysis and the mechanisms for band structure formation of SH waves in periodic ternary layered media are validated by considering a periodic ternary layered structure with the unit cell consisting of one Pb layer in the middle and two epoxy layers at the up and down sides. The thickness of the Pb layer is 10mm and that of the epoxy layers is 5mm. The material parameters of Pb and epoxy including the Young's modulus, shear modulus and mass density are 40.8187 *Pb E* GPa, 4.35005 *epoxy E* GPa, 14.9 *GPb* GPa, 1.59 *Gepoxy* GPa, 11600 *Pb*

layered medium are calculated by the formulation presented in Section 2.1. For the convenience of presentation, the results are represented by the dimensionless wavenumbers

We first consider the property of SH-wave band structures in the exemplified periodic ternary layered medium. Figure 3 gives the band structures of SH waves below 140 kHz represented as various forms of graphs. Figure 3(a) depicts the phase constant surfaces in

[ 2,2] . Figures 3(b) to 3(d) describe both the phase constant spectra in pass-bands (i.e. the

specified as 1 0.0 2 *I* , 2 0.5 2 *I* and 3 1.0 2 *I* with 1*I* , 2*I* and 3*I* being arbitrary integers. To validate our obtained results, in Fig. (3d) the phase constant spectra as / 0 *kh*

compared to the corresponding results calculated by Wang et al. (2004), and in Fig. (3e) the

) as the dimensionless wavenumber *kh* /

0 2 and 3 *qh I* /

kg/m3. The band structures of SH waves in this periodic ternary

with 0.02 *h* m being the total thickness of the unit cell, and the

and *qh* /

) and the attenuation constant spectra in stop-bands (i.e. the

1 2 are compared to their

are in the range of

is 1.0 , 0.5 and

is

when *qh* /

are

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 279

1 , which are respectively identical to those as 1 *qh I* / 02

1 2 , are the demarcations between the pass-bands and the stop-

1 2 are stop-bands. Fig. 3(e) shows the *f* - *k h* /

1 . With the increasing of *k h* /

, with the increasing of frequency the phase

0 , the first stop band is formed due

1 2 are pass-bands, whereas the

spectra obtained by our method and those

, the

appear

spectra as

spectra as

, the frequencies as

1 2 in our results are

, the demarcating

to the cutoff property of the SH waves in the constituent layers. For any *kh* /

attenuation constant spectra in form of closed loops with phase 0 and phase

. It should be noted that as *kh* /

alternately. It indicates from Figs. 3(b) to 3(e) that for any *kh* /

and the other corresponding to 3 *qh I* /

spectra corresponding to 1 *qh I* /

constant spectra and the attenuation constant spectra of characteristic SH waves in periodic ternary layered media occur alternately, i.e. the pass-bands and the stop-bands occur alternately. However, the attenuation spectra (the stop band) will advent first for any / *kh*

bands. The ranges between two adjacent bounding frequencies, with one corresponding to

ranges between two adjacent bounding frequencies with both corresponding to

frequencies of the corresponding frequency-bands rise. The first demarcation frequency

.

In Fig. 3(d), the comparison between the phase constant spectra obtained by our proposed method and those calculated by Wang et al. (2004) indicates good agreement. In Fig. 3(e),

calculated by Wang et al. (2004) also manifests close coincidence. Furthermore, the *f* -

explicitly separated, which clearly denotes the pass-bands and stop-bands. All these validate the accuracy and excellence of the proposed MRRM for dispersion characteristic

Let us now consider the formation of SH-wave band structures in the exemplified periodic ternary layered structure. We plot in Figs. (4a) and (4b) the phase and the attenuation constant spectra of characteristic SH waves together with the dispersion curves of the equivalent SH waves in the exemplified periodic ternary layered structure, as / 0.0 *kh*

curves of the equivalent SH waves and the band structures of characteristic SH waves. The fundamental real-part and the imaginary-part dispersion curves of the equivalent SH waves

real-part dispersion curve of the equivalent SH waves are attained by virtue of the zone

are obtained directly from the definition 3 22 2

, respectively, for illustrating the close relation between the dispersion

<sup>1</sup> / *se j j sj*

 

*h h ck* , while the derivative

0 2 and 3 *qh I* /

takes any other real value lie in the pass-bands between the *f* - *k h* /

Figs. 3(b) to 3(d) signify that for any *kh* /

except for / 0 *kh*

*qh* / 0 

<sup>1</sup> *qh I* / 02 

<sup>1</sup> *qh I* / 02 

*qh* / 

*k h* / 

analysis.

and / 0.5 *kh* 

folding effect.

*qh* / 0 

and those as *qh* /

or 3 *qh I* /

and those as *qh* /

increases most obviously with the rise of *k h* /

the comparison between the *f* - *k h* /

and those as 3 *qh I* /

#### 278 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

**Figure 3.** The band structures of SH waves below 140 kHz in the periodic ternary layered medium consisting of one Pb layer and two epoxy layers

It is seen from Fig. 3(a) that the phase constant surfaces in the pass-bands are symmetrical with respect to the vertical plane *kh* / 0 , which indicates that the SH waves along the positive and the negative *X* directions have identical propagation properties. This is due to the symmetry of structural configuration and material parameters of the exemplified periodic ternary layered medium with respect to *YOZ* plane. Likewise, the phase constant surfaces in Fig. 3(a) and the attenuation constant spectra in Figs. 3(b) to 3(d) are also symmetry with respect to *qh* / 0 , which indicates that the upward and downward characteristic SH waves have identical band structures. In addition, the phase constant surfaces in Fig. 3(a) are periodical with respect to *qh* / as the minimum positive period being *qh* / 2 , which indicates the periodicity of propagating characteristic SH waves along the thickness of the unit cell and reflects the zone folding effect of periodic structures. Figs. 3(b) to 3(d) signify that for any *kh* / , with the increasing of frequency the phase constant spectra and the attenuation constant spectra of characteristic SH waves in periodic ternary layered media occur alternately, i.e. the pass-bands and the stop-bands occur alternately. However, the attenuation spectra (the stop band) will advent first for any / *kh* except for / 0 *kh* . It should be noted that as *kh* / 0 , the first stop band is formed due to the cutoff property of the SH waves in the constituent layers. For any *kh* / , the attenuation constant spectra in form of closed loops with phase 0 and phase appear alternately. It indicates from Figs. 3(b) to 3(e) that for any *kh* / , the frequencies as *qh* / 0 and those as *qh* / 1 , which are respectively identical to those as 1 *qh I* / 02 and those as 3 *qh I* / 1 2 , are the demarcations between the pass-bands and the stopbands. The ranges between two adjacent bounding frequencies, with one corresponding to <sup>1</sup> *qh I* / 02 and the other corresponding to 3 *qh I* / 1 2 are pass-bands, whereas the ranges between two adjacent bounding frequencies with both corresponding to <sup>1</sup> *qh I* / 02 or 3 *qh I* / 1 2 are stop-bands. Fig. 3(e) shows the *f* - *k h* / spectra as *qh* / takes any other real value lie in the pass-bands between the *f* - *k h* / spectra as *qh* / 0 and those as *qh* / 1 . With the increasing of *k h* / , the demarcating frequencies of the corresponding frequency-bands rise. The first demarcation frequency increases most obviously with the rise of *k h* / .

278 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

**Figure 3.** The band structures of SH waves below 140 kHz in the periodic ternary layered medium

surfaces in Fig. 3(a) are periodical with respect to *qh* /

It is seen from Fig. 3(a) that the phase constant surfaces in the pass-bands are symmetrical

positive and the negative *X* directions have identical propagation properties. This is due to the symmetry of structural configuration and material parameters of the exemplified periodic ternary layered medium with respect to *YOZ* plane. Likewise, the phase constant surfaces in Fig. 3(a) and the attenuation constant spectra in Figs. 3(b) to 3(d) are also

characteristic SH waves have identical band structures. In addition, the phase constant

along the thickness of the unit cell and reflects the zone folding effect of periodic structures.

0 , which indicates that the SH waves along the

0 , which indicates that the upward and downward

as the minimum positive period

, which indicates the periodicity of propagating characteristic SH waves

consisting of one Pb layer and two epoxy layers

with respect to the vertical plane *kh* /

symmetry with respect to *qh* /

being *qh* / 2 

In Fig. 3(d), the comparison between the phase constant spectra obtained by our proposed method and those calculated by Wang et al. (2004) indicates good agreement. In Fig. 3(e), the comparison between the *f* - *k h* / spectra obtained by our method and those calculated by Wang et al. (2004) also manifests close coincidence. Furthermore, the *f k h* / spectra corresponding to 1 *qh I* / 0 2 and 3 *qh I* / 1 2 in our results are explicitly separated, which clearly denotes the pass-bands and stop-bands. All these validate the accuracy and excellence of the proposed MRRM for dispersion characteristic analysis.

Let us now consider the formation of SH-wave band structures in the exemplified periodic ternary layered structure. We plot in Figs. (4a) and (4b) the phase and the attenuation constant spectra of characteristic SH waves together with the dispersion curves of the equivalent SH waves in the exemplified periodic ternary layered structure, as / 0.0 *kh* and / 0.5 *kh* , respectively, for illustrating the close relation between the dispersion curves of the equivalent SH waves and the band structures of characteristic SH waves. The fundamental real-part and the imaginary-part dispersion curves of the equivalent SH waves are obtained directly from the definition 3 22 2 <sup>1</sup> / *se j j sj h h ck* , while the derivative real-part dispersion curve of the equivalent SH waves are attained by virtue of the zone folding effect.

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 281

0.0 , the dispersion curves of the equivalent SH waves have only

 *hm hn I* 

 

1 *h* 2 *h*

*X*

*j h*

1 2

*j*

*n*

*n h*

corresponding

( *I*

first separate with respect to frequency at the intersections 2 2 *se se*

bands, in which the attenuation constant spectra with phase 0 and phase

frequency

layered acoustic wave devices

*<sup>c</sup>*min as *kh* /

top electrode

piezoelectric propagation media

Bragg Cell

support media and substrate

bottom electrode

to the cutoff property of the SH waves in the constituent layers.

fact also applicable to SH waves in all periodic layered isotropic media.

1 2 3

*J K*

is an arbitrary integer) representing the boundaries of the Brillouin zone to give the stop-

to even numbers and odd numbers of *I* , respectively. The intermediate segments of dispersion curves between the adjacent intersections are disturbed to form the phase constant spectra in the pass-bands. Figure (4b) also shows below the minimum cutoff

the imaginary part. In this case the dispersion curve serve as the baseline during the formation of the attenuation constant spectrum in the first stop-band, which is emerged due

It should be emphasized that although the above example is a simple periodic ternary layered structure, the obtained property and formation of SH wave band structures are in

Various multi-layered acoustic wave devices with Bragg Cell can be modeled by the multilayered structures of infinite lateral extent depicted in Fig. 5, which including both nonpiezoelectric layers and piezoelectric layers. Usually, the electrodes, support layers and substrate consist of elastic (non-piezoelectric) layers. The propagation media consist of

*O*

**3. Analysis of acoustic waves in integrated multi-layered structures** 

**Figure 5.** The schematic of multi-layered structures consisting of *n* layers for modeling the multi-

*Y*

*N* 1

*N*

piezoelectric single-layer or multi-layers. The Bragg Cell can currently be made of alternate elastic layers such as W and SiO2 or alternate elastic and piezoelectric layers such as SiO2

*Z*

**Figure 4.** The band structures of characteristic SH waves and the dispersion curves of equivalent SH waves below 140 kHz in the periodic ternary layered medium consisting of one Pb layer and two epoxy layers

It is clearly seen from Figs. (4a) and (4b) that above the maximum cutoff frequency *<sup>c</sup>*max ( max min 0 *c c* while *kh* / 0.0 ), the dispersion curves of the equivalent SH waves have only the real part. In this case the fundamental and derivative dispersion curves of equivalent SH waves serve as the baselines during the band-structure formation of characteristic SH waves. In the forming process of the band structures, the dispersion curves first separate with respect to frequency at the intersections 2 2 *se se hm hn I* ( *I* is an arbitrary integer) representing the boundaries of the Brillouin zone to give the stopbands, in which the attenuation constant spectra with phase 0 and phase corresponding to even numbers and odd numbers of *I* , respectively. The intermediate segments of dispersion curves between the adjacent intersections are disturbed to form the phase constant spectra in the pass-bands. Figure (4b) also shows below the minimum cutoff frequency *<sup>c</sup>*min as *kh* / 0.0 , the dispersion curves of the equivalent SH waves have only the imaginary part. In this case the dispersion curve serve as the baseline during the formation of the attenuation constant spectrum in the first stop-band, which is emerged due to the cutoff property of the SH waves in the constituent layers.

280 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

**Figure 4.** The band structures of characteristic SH waves and the dispersion curves of equivalent SH waves below 140 kHz in the periodic ternary layered medium consisting of one Pb layer and two epoxy

only the real part. In this case the fundamental and derivative dispersion curves of equivalent SH waves serve as the baselines during the band-structure formation of characteristic SH waves. In the forming process of the band structures, the dispersion curves

0.0 ), the dispersion curves of the equivalent SH waves have

*<sup>c</sup>*max (

It is clearly seen from Figs. (4a) and (4b) that above the maximum cutoff frequency

layers

max min 0

 *c c* while *kh* / It should be emphasized that although the above example is a simple periodic ternary layered structure, the obtained property and formation of SH wave band structures are in fact also applicable to SH waves in all periodic layered isotropic media.

## **3. Analysis of acoustic waves in integrated multi-layered structures**

Various multi-layered acoustic wave devices with Bragg Cell can be modeled by the multilayered structures of infinite lateral extent depicted in Fig. 5, which including both nonpiezoelectric layers and piezoelectric layers. Usually, the electrodes, support layers and substrate consist of elastic (non-piezoelectric) layers. The propagation media consist of

**Figure 5.** The schematic of multi-layered structures consisting of *n* layers for modeling the multilayered acoustic wave devices

piezoelectric single-layer or multi-layers. The Bragg Cell can currently be made of alternate elastic layers such as W and SiO2 or alternate elastic and piezoelectric layers such as SiO2

and AlN (Lakin, 2005), and may in the future be made of alternate piezoelectric layers. Assume in the multi-layered model, each one of the *n* layers is homogeneous and the adjacent two layers are perfectly connected. To establish a general formulation for the analysis of various multi-layered acoustic wave devices with Bragg Cell, each layer in the multi-layered model is assumed as arbitrarily anisotropic. From up to down, the layers are denoted in order by numbers 1 to *n* , and the top surface, interfaces and bottom surface in turn are denoted by numbers 1 to 1 *N* ( *N n* ). Thus, the upper and lower bounding faces of an arbitrary layer *j* ( *j n* 1,2, , ) are denoted by *J* ( *J j* ) and *K* ( *K j* 1 ), respectively, and the layer *j* will also be referred to as *JK* or *KJ* . Moreover, a global coordinate system (,,) *XYZ* with its origin located on the top surface and the *Z* -axis along the thickness direction, as shown in Fig. 5, is utilized to describe the integrated multi-layered structure.

## **3.1. Modeling of the non-piezoelectric layers (electrode, Bragg Cell, support layer and substrate)**

Based on the three-dimensional linear elasticity (Stroh, 1962), the equations governing the dynamic state of a homogeneous, arbitrarily anisotropic elastic medium in absence of body forces can be written as

$$
\sigma\_{ij} = c\_{ijkl} (u\_{k,l} + u\_{l,k}) / \mathcal{Z}, \ \sigma\_{ij,j} = \rho i l\_i \tag{27}
$$

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 283

being the generalized displacement vector

*x y*

(29)

can be interpreted as the

*x y x y*

*<sup>z</sup>* **<sup>v</sup> Av** (30)

are respectively the electric displacement and the electric potential tensors,

are the piezoelectric and the permittivity constant tensors having at most 18

and 6 independent components, respectively, and all the remaining symbols have the same meanings as the corresponding ones in Eq. (27). It is seen from Eq. (28) that the coupling

Similar to the arbitrarily anisotropic elastic layer, an arbitrarily anisotropic piezoelectric layer can also be described mathematically by the state space formalism (Tarn, 2002b). In view of the global coordinate system (,,) *XYZ* in Fig. 5, the state vector is also represented

*zx zy z z D* the generalized stress vector of the first group.

**3.3. The state equations and solutions of non-piezoelectric and piezoelectric** 

ˆ( , ; ; ) e d ( , , , )e d d ,

eliminating the second group of generalized stresses, to the state equation as follows

i

*fk k z t f xyzt x y*

i ) i (i

*k x k y t*

1 1 <sup>ˆ</sup> ( , , , ) e d ( ) ( , ; ; )e d d 2 2

*f xyzt fk k z k k*

the dynamic governing equations (27) and (28) in the time-space domain can be transformed

circular frequency, *<sup>x</sup> k* and *<sup>y</sup> k* are interpreted as the wavenumbers in the *x* and *y* directions, respectively. i 1 is the unit imaginary. The *z* -dependent variable in the frequency-wavenumber domain is indicated by an over caret. Adopting the state space formalism (Tarn, 2002a, 2002b), we can reduce the transformed dynamic governing equations of a material layer corresponding to Eqs. (27) and (28) in right-handed coordinate systems, by

> d () <sup>ˆ</sup> ˆ( ) <sup>d</sup> *z*

It is noted that the transformed state vector **v**ˆ( ) *z* contains / 2 *<sup>v</sup> n* generalized displacement components and / 2 *<sup>v</sup> n* generalized stress components, with 6 *<sup>v</sup> n* and 8 *<sup>v</sup> n* for a nonpiezoelectric (elastic) layer and a piezoelectric layer, respectively. Thus, the state equation is a system of *<sup>v</sup> n* first-order ordinary differential equations. The *v v n n* coefficient matrix **A**

*z*

11 22 12 21 33 33

22 2 T1 T1

i *x y xy*

 

1 1 33 33

**G W G**

**M G G G G WG W WG** (31)

(i i ) i 2

*kx ky t*

*y x*

 

where *Di* and

as T TT [( ) ,( ) ] *<sup>u</sup>*

**layers** 

and <sup>T</sup> [, ,,] **v**

*x y*

*kij e* and *ik* 

between the mechanical and electrical fields is considered.

**vv v** , but with <sup>T</sup> [,, ,] *<sup>u</sup>* **<sup>v</sup>** *uvw*

By virtue of the triple Fourier transform pairs as follows

of the state equation can be written in a blocked form

*k k kk*

**A**

into those in the frequency-wavenumber domain. The quantity

where the comma in the subscripts and the dot above the variables imply spatial and time derivatives, *ij* and *<sup>i</sup> <sup>u</sup>* are respectively the stress and the displacement tensors, *ijkl <sup>c</sup>* denotes the elastic constant tensor having at most 21 independent components, and is the material density.

In the case of layer configuration, the state space formalism (Tarn, 2002a) can be adopted to describe mathematically the dynamic state of the medium. Referring to the global coordinate system (,,) *XYZ* in Fig. 5, we divide the stresses into two groups: the first consists of the components on the plane of constant *Z* , and the second consists of the remaining components. The combination of the displacement vector <sup>T</sup> [,, ] *<sup>u</sup>* **<sup>v</sup>** *uvw* and the

vector of first group stresses T [, ,] *zx zy z* **v** gives the state vector T TT [( ) ,( ) ] *<sup>u</sup>* **vv v** .

#### **3.2. Modeling of the piezoelectric layers (propagation media and Bragg Cell)**

According to the three-dimensional linear theory of piezoelectricity (Ding & Chen, 2001), the dynamic governing equations for the arbitrarily anisotropic piezoelectric medium in absence of both body forces and free charges are

$$\begin{cases} \sigma\_{\vec{i}\vec{j}} = \varepsilon\_{\vec{i}\vec{j}k\vec{l}} (\mu\_{k,l} + \mu\_{l,k}) / 2 + \varepsilon\_{\vec{k}\vec{i}\vec{j}} \rho\_{,k} \left\{ \sigma\_{\vec{i}\vec{j},\vec{j}} = \rho \vec{u}\_{\vec{i}} \right. \\ D\_i = \varepsilon\_{\vec{i}k\vec{l}} (\mu\_{k,l} + \mu\_{l,k}) / 2 - \beta\_{\vec{i}k} \rho\_{,k} \left. \right| \quad D\_{i,l} = 0 \end{cases} \tag{28}$$

where *Di* and are respectively the electric displacement and the electric potential tensors, *kij e* and *ik* are the piezoelectric and the permittivity constant tensors having at most 18 and 6 independent components, respectively, and all the remaining symbols have the same meanings as the corresponding ones in Eq. (27). It is seen from Eq. (28) that the coupling between the mechanical and electrical fields is considered.

Similar to the arbitrarily anisotropic elastic layer, an arbitrarily anisotropic piezoelectric layer can also be described mathematically by the state space formalism (Tarn, 2002b). In view of the global coordinate system (,,) *XYZ* in Fig. 5, the state vector is also represented as T TT [( ) ,( ) ] *<sup>u</sup>* **vv v** , but with <sup>T</sup> [,, ,] *<sup>u</sup>* **<sup>v</sup>** *uvw* being the generalized displacement vector and <sup>T</sup> [, ,,] **v** *zx zy z z D* the generalized stress vector of the first group.

## **3.3. The state equations and solutions of non-piezoelectric and piezoelectric layers**

By virtue of the triple Fourier transform pairs as follows

282 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

utilized to describe the integrated multi-layered structure.

**layer and substrate)** 

forces can be written as

vector of first group stresses T [, ,]

absence of both body forces and free charges are

  *zx zy z* **v** 

derivatives, *ij*

material density.

and AlN (Lakin, 2005), and may in the future be made of alternate piezoelectric layers. Assume in the multi-layered model, each one of the *n* layers is homogeneous and the adjacent two layers are perfectly connected. To establish a general formulation for the analysis of various multi-layered acoustic wave devices with Bragg Cell, each layer in the multi-layered model is assumed as arbitrarily anisotropic. From up to down, the layers are denoted in order by numbers 1 to *n* , and the top surface, interfaces and bottom surface in turn are denoted by numbers 1 to 1 *N* ( *N n* ). Thus, the upper and lower bounding faces of an arbitrary layer *j* ( *j n* 1,2, , ) are denoted by *J* ( *J j* ) and *K* ( *K j* 1 ), respectively, and the layer *j* will also be referred to as *JK* or *KJ* . Moreover, a global coordinate system (,,) *XYZ* with its origin located on the top surface and the *Z* -axis along the thickness direction, as shown in Fig. 5, is

**3.1. Modeling of the non-piezoelectric layers (electrode, Bragg Cell, support** 

Based on the three-dimensional linear elasticity (Stroh, 1962), the equations governing the dynamic state of a homogeneous, arbitrarily anisotropic elastic medium in absence of body

,, , ( ) / 2, *ij ijkl k l l k ij j i*

where the comma in the subscripts and the dot above the variables imply spatial and time

In the case of layer configuration, the state space formalism (Tarn, 2002a) can be adopted to describe mathematically the dynamic state of the medium. Referring to the global coordinate system (,,) *XYZ* in Fig. 5, we divide the stresses into two groups: the first consists of the components on the plane of constant *Z* , and the second consists of the remaining components. The combination of the displacement vector <sup>T</sup> [,, ] *<sup>u</sup>* **<sup>v</sup>** *uvw* and the

**3.2. Modeling of the piezoelectric layers (propagation media and Bragg Cell)** 

*D eu u D*

 

According to the three-dimensional linear theory of piezoelectricity (Ding & Chen, 2001), the dynamic governing equations for the arbitrarily anisotropic piezoelectric medium in

> ,, , , ,, , , ( )/2 , ( )/2 <sup>0</sup> *ij ijkl k l l k kij k ij j i i ikl k l l k ik k i i*

 

 

*cu u e u*

and *<sup>i</sup> <sup>u</sup>* are respectively the stress and the displacement tensors, *ijkl <sup>c</sup>* denotes

 

gives the state vector T TT [( ) ,( ) ] *<sup>u</sup>*

*u* (27)

**vv v** .

(28)

is the

*cu u*

the elastic constant tensor having at most 21 independent components, and

$$\begin{split} \hat{f}(k\_{\boldsymbol{x}},k\_{\boldsymbol{y}};z;o) &= \int\_{-\alpha}^{+\alpha} \mathrm{e}^{-i\alpha t} \, \mathrm{d}t \int\_{-\alpha}^{+\alpha} \int\_{-\alpha}^{+\alpha} f(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z},t) \mathrm{e}^{(\boldsymbol{k}\_{\boldsymbol{x}}\boldsymbol{x}+\boldsymbol{k}\_{\boldsymbol{y}}\boldsymbol{y})} \, \mathrm{d}\boldsymbol{x} \, \mathrm{d}\boldsymbol{y} \\ f(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z},t) &= \int\_{-\alpha}^{+\alpha} \frac{1}{2\pi} \mathrm{e}^{i\alpha t} \, \mathrm{d}\boldsymbol{o} \int\_{-\alpha}^{+\alpha} \int\_{-\alpha}^{+\alpha} (\frac{1}{2\pi})^2 \hat{f}(\boldsymbol{k}\_{\boldsymbol{x}},\boldsymbol{k}\_{\boldsymbol{y}};\boldsymbol{z};o) \mathrm{e}^{-(\boldsymbol{k}\_{\boldsymbol{x}}\boldsymbol{x}+\boldsymbol{k}\_{\boldsymbol{y}}\boldsymbol{y})} \, \mathrm{d}\boldsymbol{k}\_{\boldsymbol{x}} \, \mathrm{d}\boldsymbol{k}\_{\boldsymbol{y}} \end{split} \tag{29}$$

the dynamic governing equations (27) and (28) in the time-space domain can be transformed into those in the frequency-wavenumber domain. The quantity can be interpreted as the circular frequency, *<sup>x</sup> k* and *<sup>y</sup> k* are interpreted as the wavenumbers in the *x* and *y* directions, respectively. i 1 is the unit imaginary. The *z* -dependent variable in the frequency-wavenumber domain is indicated by an over caret. Adopting the state space formalism (Tarn, 2002a, 2002b), we can reduce the transformed dynamic governing equations of a material layer corresponding to Eqs. (27) and (28) in right-handed coordinate systems, by eliminating the second group of generalized stresses, to the state equation as follows

$$\frac{\text{d}\,\hat{\mathbf{v}}(z)}{\text{d}\,z} = \mathbf{A}\hat{\mathbf{v}}(z) \tag{30}$$

It is noted that the transformed state vector **v**ˆ( ) *z* contains / 2 *<sup>v</sup> n* generalized displacement components and / 2 *<sup>v</sup> n* generalized stress components, with 6 *<sup>v</sup> n* and 8 *<sup>v</sup> n* for a nonpiezoelectric (elastic) layer and a piezoelectric layer, respectively. Thus, the state equation is a system of *<sup>v</sup> n* first-order ordinary differential equations. The *v v n n* coefficient matrix **A** of the state equation can be written in a blocked form

$$\mathbf{A} = \begin{vmatrix} -\mathbf{i}\mathbf{G}\_{33}^{-1}\mathbf{W} & \mathbf{G}\_{33}^{-1} \\ -\rho\rho^2\mathbf{M} + k\_x^2\mathbf{G}\_{11} + k\_y^2\mathbf{G}\_{22} + k\_xk\_y\left(\mathbf{G}\_{12} + \mathbf{G}\_{21}\right) - \mathbf{W}^T\mathbf{G}\_{33}^{-1}\mathbf{W} & -\mathbf{i}\mathbf{W}^T\mathbf{G}\_{33}^{-1} \end{vmatrix} \tag{31}$$

where 31 32 ( ) *x y* **W GG** *k k* . Assuming that the correspondence between the digital and coordinate indices follows 1 *x* , 2 *y* and 3 *z* , we have

$$\mathbf{G}\_{kl} = \begin{bmatrix} \mathbf{c}\_{1k1l} & \mathbf{c}\_{1k2l} & \mathbf{c}\_{1k3l} \\ \mathbf{c}\_{2k1l} & \mathbf{c}\_{2k2l} & \mathbf{c}\_{2k3l} \\ \mathbf{c}\_{3k1l} & \mathbf{c}\_{3k2l} & \mathbf{c}\_{3k3l} \end{bmatrix}, \ \mathbf{M} = \mathbf{I}\_3 \tag{32}$$

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 285

**3.4. Reverberation-ray matrix analysis of integrated multi-layered structures** 

**Figure 6.** Description of a typical layer *j* within the multi-layered model in local dual coordinates

*JK x*

*X*

*JK* **a**

*KJ* **a**

*JK <sup>z</sup> JK <sup>y</sup>*

*<sup>Y</sup> <sup>Z</sup>*

*O*

*KJ z*

*y k*

It is seen from Fig. 6 that the local dual coordinates are both right-handed, thus the state equations for an arbitrary layer *j* (i.e. *JK* or *KJ* ) in (,,) *JK JK JK xyz* and (,,) *KJ KJ KJ xyz* have the same form as Eq. (30). The traveling wave solutions to them can be written according to

<sup>ˆ</sup> ( ) exp( ) exp( ) , <sup>ˆ</sup> ( ) exp( ) *JK JK JK JK JK JK JK JK*

 

 

*j*

*j h*

*x k*

*JK* **d**

*KJ* **d**

(35)

. (36)

*JK JK JK JK JK JK JK JK*

*KJ KJ KJ KJ KJ KJ KJ KJ*

 **<sup>v</sup> <sup>Φ</sup> ΦΦ Λ <sup>0</sup> <sup>a</sup> <sup>Λ</sup> <sup>w</sup> vΦ Φ Φ 0 Λ d**

 **<sup>v</sup> <sup>Φ</sup> Φ Φ <sup>Λ</sup> <sup>0</sup> <sup>a</sup> <sup>Λ</sup> <sup>w</sup> vΦ Φ Φ 0 Λ d**

*z z*

<sup>ˆ</sup> ( ) exp( ) exp( ) <sup>ˆ</sup> ( ) exp( ) *KJ KJ KJ KJ KJ KJ KJ KJ*

*z z*

*3.4.2. Traveling wave solutions to the state variables* 

*KJ x KJ y*

*K*

( 1) *j*

*J* ( )*j*

*u u JK JK JK u u*

*u u KJ KJ KJ u u*

 

  *z*

*z*

*z z*

*z z*

Eq. (34) as follows

Within the framework of MRRM, the physical variables associated with any surface/interface *J* ( *J N* 1,2, , 1 ) will be described in the global coordinate system (,,) *XYZ* as shown in Fig. 5 for the convenience of system analysis, and will be affixed with single superscript *J* to indicate their affiliation. The physical variables associated with any layer *j* (i.e. *JK* or *KJ* , *j n* 1,2, , ) will be described in the local dual coordinates (,,) *JK JK JK xyz* or (,,) *KJ KJ KJ xyz* as shown in Fig. 6 for the sake of member analysis, and will be affixed with double superscripts *JK* or *KJ* to denote the corresponding coordinate system and the pertaining layer. To make the sign convection clear, physical variables are deemed to be positive as it is along the positive direction of the pertinent coordinate axis.

*3.4.1. Description of the structural system* 

for a layer of arbitrarily anisotropic elastic material, and

$$\mathbf{G}\_{kl} = \begin{bmatrix} c\_{1k1l} & c\_{1k2l} & c\_{1k3l} & e\_{1lk} \\ c\_{2k1l} & c\_{2k2l} & c\_{2k3l} & e\_{12k} \\ c\_{3k1l} & c\_{3k2l} & c\_{3k3l} & e\_{13k} \\ c\_{k1l} & c\_{k2l} & e\_{k3l} & -\mathcal{B}\_{kl} \end{bmatrix}, \mathbf{M} = \begin{bmatrix} \mathbf{I}\_3 & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \tag{33}$$

for a layer of arbitrarily anisotropic piezoelectric material, with 3**I** the identity matrix of order 3.

According to the theory of ordinary differential equation (Coddington & Levinson, 1955), the solution to the state equation (30) can be expressed, in a form of traveling waves, as

$$\begin{aligned} \hat{\mathbf{v}}(z) &= \boldsymbol{\Phi} \exp(\boldsymbol{\Lambda}z) \mathbf{w} = \begin{bmatrix} \boldsymbol{\Phi}\_{-} & \boldsymbol{\Phi}\_{+} \end{bmatrix} \begin{bmatrix} \exp(\boldsymbol{\Lambda}\_{-}z) & \mathbf{0} \\ \mathbf{0} & \exp(\boldsymbol{\Lambda}\_{+}z) \end{bmatrix} \begin{bmatrix} \mathbf{a} \\ \mathbf{d} \end{bmatrix} \\ &= \begin{bmatrix} \boldsymbol{\Psi}\_{u}(z) \\ \hat{\mathbf{v}}\_{\sigma}(z) \end{bmatrix} = \begin{bmatrix} \boldsymbol{\Phi}\_{u} \\ \boldsymbol{\Phi}\_{\sigma} \end{bmatrix} \exp(\boldsymbol{\Lambda}z) \mathbf{w} = \begin{bmatrix} \boldsymbol{\Phi}\_{u-} & \boldsymbol{\Phi}\_{u+} \\ \boldsymbol{\Phi}\_{\sigma-} & \boldsymbol{\Phi}\_{\sigma+} \end{bmatrix} \begin{bmatrix} \exp(\boldsymbol{\Lambda}\_{-}z) & \mathbf{0} \\ \mathbf{0} & \exp(\boldsymbol{\Lambda}\_{+}z) \end{bmatrix} \begin{bmatrix} \mathbf{a} \\ \mathbf{d} \end{bmatrix} \end{aligned} \tag{34}$$

where exp( ) denotes the matrix exponential function, **Λ** and **Φ** are respectively the *v v n n* diagonal eigenvalue matrix and square eigenvector matrix of the coefficient matrix **A** , **w** is the vector of undetermined wave amplitudes with *<sup>v</sup> n* components. **Λ** ( *a a n n* ) and **Λ** ( *d d n n* ) are the diagonal sub-matrices of **Λ** corresponding respectively to the arriving wave vector **a** with *<sup>a</sup> n* wave amplitudes and the departing wave vector **d** with *<sup>d</sup> n* wave amplitudes. **Φ** ( *v a n n* ) and **Φ** ( *v d n n* ) are the corresponding sub matrices of **Φ** . **Φ***u* and **Φ** are the / 2 *v v n n* sub-matrices of **Φ** corresponding to the generalized displacement and stress vectors, respectively. The sub-matrices **Φ***u* , **Φ** , **Φ***u* and **Φ** are defined accordingly. It should be noted that **a** consists of those wave amplitudes *wi* in **w** , which correspond to the eigenvalues *<sup>i</sup>* satisfying Re( ) 0 *<sup>i</sup>* or Re( ) 0,Im( ) 0 *i i* , and the remaining wave amplitudes in **w** form **d** . Obviously, we always have T TT **w ad** [, ] and *adv nnn* .

#### **3.4. Reverberation-ray matrix analysis of integrated multi-layered structures**

#### *3.4.1. Description of the structural system*

284 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

coordinate indices follows 1 *x* , 2 *y* and 3 *z* , we have

for a layer of arbitrarily anisotropic elastic material, and

*kl*

exp ˆ( ) exp

**v Φ Λ w Φ Φ**

 

*z z*

 

always have T TT **w ad** [, ] and *adv nnn* .

**Φ** . **Φ***u* and **Φ**

Re( ) 0,Im( ) 0 *i i*

order 3.

where 31 32 ( ) *x y* **W GG** *k k* . Assuming that the correspondence between the digital and

11 12 13

 

*kl k l k l k l*

11 12 13 1

*ccc e ccc e ccc e eee*

*kl kl kl lk kl kl kl lk*

31 32 33 3 123

*u u u u*

ˆ ( ) exp

*z*

displacement and stress vectors, respectively. The sub-matrices **Φ***u* , **Φ**

**w** , which correspond to the eigenvalues *<sup>i</sup>*

*z z*

*kl kl kl lk k l k l k l kl*

*ccc ccc ccc*

*kl kl kl*

31 32 33

*kl kl kl*

21 22 23 2 3

for a layer of arbitrarily anisotropic piezoelectric material, with 3**I** the identity matrix of

According to the theory of ordinary differential equation (Coddington & Levinson, 1955), the solution to the state equation (30) can be expressed, in a form of traveling waves, as

 

exp , <sup>ˆ</sup> ( ) exp

*z z*

**<sup>v</sup> <sup>Φ</sup> Φ Φ <sup>Λ</sup> <sup>0</sup> <sup>a</sup> <sup>Λ</sup> <sup>w</sup> v Φ Φ Φ 0 Λ d**

where exp( ) denotes the matrix exponential function, **Λ** and **Φ** are respectively the *v v n n* diagonal eigenvalue matrix and square eigenvector matrix of the coefficient matrix **A** , **w** is the vector of undetermined wave amplitudes with *<sup>v</sup> n* components. **Λ** ( *a a n n* ) and **Λ** ( *d d n n* ) are the diagonal sub-matrices of **Λ** corresponding respectively to the arriving wave vector **a** with *<sup>a</sup> n* wave amplitudes and the departing wave vector **d** with *<sup>d</sup> n* wave amplitudes. **Φ** ( *v a n n* ) and **Φ** ( *v d n n* ) are the corresponding sub matrices of

are defined accordingly. It should be noted that **a** consists of those wave amplitudes *wi* in

*z*

21 22 23 3

,

**G M I** (32)

<sup>0</sup> , 0 0

**<sup>I</sup> G M** (33)

*z*

exp

are the / 2 *v v n n* sub-matrices of **Φ** corresponding to the generalized

, and the remaining wave amplitudes in **w** form **d** . Obviously, we

**Λ 0 a**

**0 Λ d**

satisfying Re( ) 0 *<sup>i</sup>*

, **Φ***u* and **Φ**

or

(34)

Within the framework of MRRM, the physical variables associated with any surface/interface *J* ( *J N* 1,2, , 1 ) will be described in the global coordinate system (,,) *XYZ* as shown in Fig. 5 for the convenience of system analysis, and will be affixed with single superscript *J* to indicate their affiliation. The physical variables associated with any layer *j* (i.e. *JK* or *KJ* , *j n* 1,2, , ) will be described in the local dual coordinates (,,) *JK JK JK xyz* or (,,) *KJ KJ KJ xyz* as shown in Fig. 6 for the sake of member analysis, and will be affixed with double superscripts *JK* or *KJ* to denote the corresponding coordinate system and the pertaining layer. To make the sign convection clear, physical variables are deemed to be positive as it is along the positive direction of the pertinent coordinate axis.

**Figure 6.** Description of a typical layer *j* within the multi-layered model in local dual coordinates

#### *3.4.2. Traveling wave solutions to the state variables*

It is seen from Fig. 6 that the local dual coordinates are both right-handed, thus the state equations for an arbitrary layer *j* (i.e. *JK* or *KJ* ) in (,,) *JK JK JK xyz* and (,,) *KJ KJ KJ xyz* have the same form as Eq. (30). The traveling wave solutions to them can be written according to Eq. (34) as follows

<sup>ˆ</sup> ( ) exp( ) exp( ) , <sup>ˆ</sup> ( ) exp( ) *JK JK JK JK JK JK JK JK u u JK JK JK u u JK JK JK JK JK JK JK JK z z z z z* **<sup>v</sup> <sup>Φ</sup> Φ Φ <sup>Λ</sup> <sup>0</sup> <sup>a</sup> <sup>Λ</sup> <sup>w</sup> v Φ Φ Φ 0 Λ d** (35) <sup>ˆ</sup> ( ) exp( ) exp( ) <sup>ˆ</sup> ( ) exp( ) *KJ KJ KJ KJ KJ KJ KJ KJ u u KJ KJ KJ u u KJ KJ KJ KJ KJ KJ KJ KJ z z z z z* **<sup>v</sup> <sup>Φ</sup> ΦΦ Λ <sup>0</sup> <sup>a</sup> <sup>Λ</sup> <sup>w</sup> v Φ Φ Φ 0 Λ d** . (36)

#### *3.4.3. Scattering relations from coupling conditions on surfaces and at interfaces*

Consider the compatibility of generalized displacements and the equilibrium of generalized stresses on surfaces and at interfaces. The spectral coupling equations on the top surface 1 , at any interface *J* and on the bottom surface 1 *N* are expressed respectively as

$$
\hat{\mathbf{v}}\_{\
u}^{12}(\mathbf{0}) = \hat{\mathbf{v}}\_{\
u \to \prime}^{1} \quad \hat{\mathbf{v}}\_{\sigma}^{12}(\mathbf{0}) + \hat{\mathbf{v}}\_{\sigma \to}^{1} = \mathbf{0} \tag{37}
$$

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 287

<sup>0</sup> **Aa Dd s** , (43)

*K* **T**

(44)

*<sup>K</sup>* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup> K* **v** .

surface 1 *<sup>N</sup>* , respectively, the corresponding coefficient matrices <sup>1</sup> **<sup>T</sup>***<sup>K</sup>* , *<sup>J</sup>* **<sup>T</sup>***K* and *<sup>N</sup>* <sup>1</sup>

The local scattering relations of top surface, interfaces and bottom surface are grouped

<sup>T</sup> 12 T 21 T 23 T T T ( 1) T ( 1) T

( ) ,( ) ,( ) , ,( ) ,( ) , ,( ) ,( )

**aa a a a a a a**

**dd d d d d d d**

0 00 0 0 ( ) ,( ) , ,( ) , ,( ) *J N* **s ss s s** is the global excitation source vector.

coordinates. This is the main advantage of introducing the local dual coordinates.

It is noticed that the exponential functions in the solutions to the state variables of layers as shown in Eqs. (35) and (36) disappear in the scattering relations, since the thickness coordinates on the surfaces and at the interfaces are always zero in the corresponding local

Considering the compatibility between generalized displacements (generalized stresses) represented in local coordinates (,,) *JK JK JK xyz* and the corresponding ones represented in

> ˆˆ ˆ ˆ ( ) ( ), ( ) ( ) *JK JK KJ JK KJ JK JK KJ JK KJ <sup>u</sup> u u z hz z hz*

where *JK h* ( *KJ h* ) denotes the thickness of layer *JK* (*KJ*). Substitution of Eqs. (35) and (36)

*JK JK JK KJ JK JK JK*

 **a Λ 0 d P0 d 0 I**

*KJ JK JK JK KJ JK KJ*

**<sup>a</sup> <sup>0</sup> <sup>Λ</sup> d 0P d I 0** (47)

 

*<sup>v</sup>* **<sup>Φ</sup> <sup>T</sup> <sup>Φ</sup>** , <sup>1</sup>

*v v*

*a*

*d*

**T T** , *JK KJ*

*a d n n* and

**v Tv v Tv** (46)

<sup>T</sup> 12 T 21 T 23 T T T ( 1) T ( 1) T ( ) ,( ) ,( ) , ,( ) ,( ) , ,( ) ,( ) ,

*JI JK N N N N*

1 2 <sup>1</sup> 1 2 <sup>1</sup> , ,, ,, , , ,, ,, *J N J N* **A AA A A D DD D D** (45)

*JI JK N N N N*

**T TT** in accordance with <sup>1</sup> <sup>ˆ</sup> *<sup>K</sup>* **<sup>v</sup>** , ˆ *<sup>J</sup>*

consist of the components from , *v u*

together from up to down to give the global scattering relation

the corresponding coefficient matrices **A** and **D** are

<sup>T</sup> 1 T 2 T T ( 1) T

*3.4.4. Phase relations from compatibility conditions of layers* 

into Eq. (46), one obtains, by noticing *JK KJ* **Λ Λ** , *JK KJ*

*d a n n* , the local phase relation of a typical layer *j* (i.e. *JK* or *KJ*)

exp( )

*h*

(,,) *KJ KJ KJ xyz* of any layer *j* (i.e. *JK* or *KJ*), we have

exp( )

*h*

and

*JK KJ*

where the global arriving and departing wave vectors **a** and **d** are

$$\mathbf{T}\_{\boldsymbol{u}}\hat{\mathbf{v}}\_{\boldsymbol{u}}^{\mathrm{II}}(\mathbf{0}) = \hat{\mathbf{v}}\_{\boldsymbol{u}}^{\mathrm{IK}}(\mathbf{0}) = \hat{\mathbf{v}}\_{\boldsymbol{u}\boldsymbol{E}\prime}^{\mathrm{I}\prime} \quad \mathbf{T}\_{\boldsymbol{\sigma}}\hat{\mathbf{v}}\_{\boldsymbol{\sigma}}^{\mathrm{II}}(\mathbf{0}) + \hat{\mathbf{v}}\_{\boldsymbol{\sigma}}^{\mathrm{IK}}(\mathbf{0}) + \hat{\mathbf{v}}\_{\boldsymbol{\sigma}\boldsymbol{E}}^{\mathrm{I}} = \mathbf{0} \tag{38}$$

$$\mathbf{T}\_{\mu}\hat{\mathbf{v}}\_{\mu}^{(N+1)N}(0) = \hat{\mathbf{v}}\_{\mu E}^{(N+1)}, \quad \mathbf{T}\_{\sigma}\hat{\mathbf{v}}\_{\sigma}^{(N+1)N}(0) + \hat{\mathbf{v}}\_{\sigma E}^{(N+1)} = \mathbf{0} \tag{39}$$

where <sup>1</sup> <sup>ˆ</sup> *uE* **<sup>v</sup>** , ˆ *<sup>J</sup> uE* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup> uE* **v** are the generalized displacement vectors of top surface 1 , interface *J* and bottom surface 1 *N* , respectively, <sup>1</sup> ˆ *<sup>E</sup>* **<sup>v</sup>** , ˆ *<sup>J</sup> <sup>E</sup>* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup> E* **v** are the corresponding generalized stress vectors, *<sup>u</sup>* **T T** are the coordinate transformation matrix that equal to 1,1, 1 for non-piezoelectric (elastic) layers and 1,1, 1, 1 for piezoelectric layers. Here and after denotes the (block) diagonal matrix with elements (or sub-matrices) only on the main diagonal.

It should be noticed that halves of all the components in vectors <sup>1</sup> <sup>ˆ</sup> *uE* **<sup>v</sup>** and <sup>1</sup> <sup>ˆ</sup> *<sup>E</sup>* **v** , in vectors ˆ *J uE* **<sup>v</sup>** and ˆ *<sup>J</sup> <sup>E</sup>* **<sup>v</sup>** , and in vectors ( 1) <sup>ˆ</sup> *<sup>N</sup> uE* **v** and ( 1) ˆ *<sup>N</sup> E* **v** are known, which are denoted by vectors <sup>1</sup> <sup>ˆ</sup> *<sup>K</sup>* **<sup>v</sup>**, ˆ *<sup>J</sup> <sup>K</sup>* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup> K* **v** , respectively. Substituting the solutions to the state variables of layers as given in Eqs. (35) and (36) into those coupling equations containing <sup>1</sup> <sup>ˆ</sup> *<sup>K</sup>* **<sup>v</sup>** , ˆ *<sup>J</sup> <sup>K</sup>* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup> K* **v** , we can obtain respectively the local scattering relations of top surface 1 , interface *J* and bottom surface 1 *N* as follows

$$\mathbf{A}^1 \mathbf{a}^1 + \mathbf{D}^1 \mathbf{d}^1 = \mathbf{T}\_K^1 \hat{\mathbf{v}}\_K^1 = \mathbf{s}\_0^1 \tag{40}$$

$$\mathbf{A}^{J}\mathbf{a}^{J} + \mathbf{D}^{J}\mathbf{d}^{J} = \mathbf{T}\_{K}^{J}\hat{\mathbf{v}}\_{K}^{J} = \mathbf{s}\_{0}^{J} \tag{41}$$

$$\mathbf{A}^{N+1}\mathbf{a}^{N+1} + \mathbf{D}^{N+1}\mathbf{d}^{N+1} = \mathbf{T}\_K^{N+1}\hat{\mathbf{v}}\_K^{N+1} = \mathbf{s}\_0^{N+1} \tag{42}$$

where 1 12 **a a** ( 1 12 **d d** ), T TT [( ) ,( ) ] *J JI JK* **aa a** ( T TT [( ) ,( ) ] *J JI JK* **dd d** ) and *N NN* 1 ( 1) **a a** ( *N NN* 1 ( 1) **d d** ) are the arriving (departing) wave vectors of top surface 1 , interface *J* and bottom surface 1 *N* , respectively, the corresponding coefficient matrices <sup>1</sup> **A** ( <sup>1</sup> **D** ), *<sup>J</sup>* **A** ( *<sup>J</sup>* **<sup>D</sup>** ) and *<sup>N</sup>*<sup>1</sup> **<sup>A</sup>** ( *<sup>N</sup>*<sup>1</sup> **<sup>D</sup>** ) have components extracted, in accordance with <sup>1</sup> <sup>ˆ</sup> *<sup>K</sup>* **<sup>v</sup>** , ˆ *<sup>J</sup> <sup>K</sup>* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup> K* **v** , from respectively <sup>12</sup> **<sup>Φ</sup>** ( <sup>12</sup> **<sup>Φ</sup>** ), [ ,] *JI JK* **<sup>T</sup>***v***Φ Φ** ([ ,] *JI JK* **<sup>T</sup>***v***Φ Φ** ) and ( 1) *N N v* **<sup>T</sup> <sup>Φ</sup>** ( ( 1) *N N v* **<sup>T</sup> <sup>Φ</sup>** ), <sup>1</sup> <sup>0</sup> **s** , 0 *<sup>J</sup>* **s** and ( 1) 0 *<sup>N</sup>* **s** are the excitation source vectors of top surface 1 , interface *J* and bottom surface 1 *<sup>N</sup>* , respectively, the corresponding coefficient matrices <sup>1</sup> **<sup>T</sup>***<sup>K</sup>* , *<sup>J</sup>* **<sup>T</sup>***K* and *<sup>N</sup>* <sup>1</sup> *K* **T** consist of the components from , *v u* **T TT** in accordance with <sup>1</sup> <sup>ˆ</sup> *<sup>K</sup>* **<sup>v</sup>** , ˆ *<sup>J</sup> <sup>K</sup>* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup> K* **v** .

The local scattering relations of top surface, interfaces and bottom surface are grouped together from up to down to give the global scattering relation

$$\mathbf{A}\mathbf{a} + \mathbf{D}\mathbf{d} = \mathbf{s}\_{0^{\prime}} \tag{43}$$

where the global arriving and departing wave vectors **a** and **d** are

$$\mathbf{a} = \left[ (\mathbf{a}^{12})^{\mathrm{T}}, (\mathbf{a}^{21})^{\mathrm{T}}, (\mathbf{a}^{22})^{\mathrm{T}}, \cdots, (\mathbf{a}^{\mathrm{Il}})^{\mathrm{T}}, (\mathbf{a}^{\mathrm{IK}})^{\mathrm{T}}, \cdots, (\mathbf{a}^{\mathrm{N(N+1)}})^{\mathrm{T}}, (\mathbf{a}^{(\mathrm{N+1})N})^{\mathrm{T}} \right]^{\mathrm{T}},$$

$$\mathbf{d} = \left[ (\mathbf{d}^{12})^{\mathrm{T}}, (\mathbf{d}^{21})^{\mathrm{T}}, (\mathbf{d}^{23})^{\mathrm{T}}, \cdots, (\mathbf{d}^{\mathrm{Il}})^{\mathrm{T}}, (\mathbf{d}^{\mathrm{IK}})^{\mathrm{T}}, \cdots, (\mathbf{d}^{\mathrm{N(N+1)}})^{\mathrm{T}}, (\mathbf{d}^{(\mathrm{N+1})N})^{\mathrm{T}} \right]^{\mathrm{T}} \tag{44}$$

the corresponding coefficient matrices **A** and **D** are

286 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

*u u u uE*

*u u uE*

interface *J* and bottom surface 1 *N* , respectively, <sup>1</sup> ˆ

*uE* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup> uE*

corresponding generalized stress vectors, *<sup>u</sup>*

(or sub-matrices) only on the main diagonal.

*<sup>E</sup>* **<sup>v</sup>** , and in vectors ( 1) <sup>ˆ</sup> *<sup>N</sup>*

where <sup>1</sup> <sup>ˆ</sup> *uE* **<sup>v</sup>** , ˆ *<sup>J</sup>*

*<sup>K</sup>* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup> K*

bottom surface 1 *N* as follows

ˆ *J uE* **<sup>v</sup>** and ˆ *<sup>J</sup>*

0

*<sup>J</sup>* **s** and ( 1) 0

<sup>1</sup> <sup>ˆ</sup> *<sup>K</sup>* **<sup>v</sup>** , ˆ *<sup>J</sup>*

*3.4.3. Scattering relations from coupling conditions on surfaces and at interfaces* 

at any interface *J* and on the bottom surface 1 *N* are expressed respectively as

12 1 12 1 ˆ ˆˆ ˆ (0) , (0) *u uE* 

ˆˆ ˆˆˆ ˆ (0) (0) , (0) (0) *JI JK J JI JK J*

( 1) ( 1) ( 1) ( 1) ˆ ˆˆ ˆ (0) , (0) *NN N NN N*

 

**v** , respectively. Substituting the solutions to the state variables of layers as

that equal to 1,1, 1 for non-piezoelectric (elastic) layers and 1,1, 1, 1 for piezoelectric layers. Here and after denotes the (block) diagonal matrix with elements

> *E*

can obtain respectively the local scattering relations of top surface 1 , interface *J* and

11 1 1 1 1 1

<sup>0</sup> <sup>ˆ</sup> *JJ J J J J J*

11 1 1 1 1 1 <sup>0</sup> <sup>ˆ</sup> *NN N N N N N*

*<sup>N</sup>* **s** are the excitation source vectors of top surface 1 , interface *J* and bottom

where 1 12 **a a** ( 1 12 **d d** ), T TT [( ) ,( ) ] *J JI JK* **aa a** ( T TT [( ) ,( ) ] *J JI JK* **dd d** ) and *N NN* 1 ( 1) **a a** ( *N NN* 1 ( 1) **d d** ) are the arriving (departing) wave vectors of top surface 1 , interface *J* and bottom surface 1 *N* , respectively, the corresponding coefficient matrices <sup>1</sup> **A** ( <sup>1</sup> **D** ), *<sup>J</sup>* **A**

( *<sup>J</sup>* **<sup>D</sup>** ) and *<sup>N</sup>*<sup>1</sup> **<sup>A</sup>** ( *<sup>N</sup>*<sup>1</sup> **<sup>D</sup>** ) have components extracted, in accordance with <sup>1</sup> <sup>ˆ</sup> *<sup>K</sup>* **<sup>v</sup>** , ˆ *<sup>J</sup>*

from respectively <sup>12</sup> **<sup>Φ</sup>** ( <sup>12</sup> **<sup>Φ</sup>** ), [ ,] *JI JK* **<sup>T</sup>***v***Φ Φ** ([ ,] *JI JK* **<sup>T</sup>***v***Φ Φ** ) and ( 1) *N N*

*K K*

It should be noticed that halves of all the components in vectors <sup>1</sup> <sup>ˆ</sup> *uE* **<sup>v</sup>** and <sup>1</sup> <sup>ˆ</sup>

**v** and ( 1) ˆ *<sup>N</sup>*

given in Eqs. (35) and (36) into those coupling equations containing <sup>1</sup> <sup>ˆ</sup> *<sup>K</sup>* **<sup>v</sup>** , ˆ *<sup>J</sup>*

*uE*

 

> 

*<sup>E</sup>* **Tv v v Tv v v 0** (38)

**Tv v Tv v 0** (39)

**v** are the generalized displacement vectors of top surface 1 ,

*<sup>E</sup>* **v vv v0** (37)

*E*

 *<sup>E</sup>* **<sup>v</sup>** , ˆ *<sup>J</sup>* 

 

**T T** are the coordinate transformation matrix

**v** are known, which are denoted by vectors

<sup>0</sup> ˆ *K K* **Aa Dd T v s** (40)

*K K* **Aa Dd T v s** (41)

*v*

 **<sup>T</sup> <sup>Φ</sup>** ( ( 1) *N N v*

**Aa Dd Tv s** (42)

*<sup>E</sup>* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup>*

*E*

*<sup>K</sup>* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup> K* **v** , we

**v** are the

*<sup>E</sup>* **v** , in vectors

*<sup>K</sup>* **<sup>v</sup>** and ( 1) <sup>ˆ</sup> *<sup>N</sup> K* **v** ,

**<sup>T</sup> <sup>Φ</sup>** ), <sup>1</sup>

<sup>0</sup> **s** ,

Consider the compatibility of generalized displacements and the equilibrium of generalized stresses on surfaces and at interfaces. The spectral coupling equations on the top surface 1 ,

$$\mathbf{A} = \mathbf{A}^1, \mathbf{A}^2, \dots, \mathbf{A}^J, \dots, \mathbf{A}^{N+1} >\_{\prime} \mathbf{D} = \mathbf{C}^1, \mathbf{D}^2, \dots, \mathbf{D}^J, \dots, \mathbf{D}^{N+1} > \tag{45}$$

and <sup>T</sup> 1 T 2 T T ( 1) T 0 00 0 0 ( ) ,( ) , ,( ) , ,( ) *J N* **s ss s s** is the global excitation source vector.

It is noticed that the exponential functions in the solutions to the state variables of layers as shown in Eqs. (35) and (36) disappear in the scattering relations, since the thickness coordinates on the surfaces and at the interfaces are always zero in the corresponding local coordinates. This is the main advantage of introducing the local dual coordinates.

#### *3.4.4. Phase relations from compatibility conditions of layers*

Considering the compatibility between generalized displacements (generalized stresses) represented in local coordinates (,,) *JK JK JK xyz* and the corresponding ones represented in (,,) *KJ KJ KJ xyz* of any layer *j* (i.e. *JK* or *KJ*), we have

$$
\hat{\mathbf{v}}\_{\boldsymbol{\mu}}^{\rm IK}(\boldsymbol{z}^{\rm IK}) = \mathbf{T}\_{\boldsymbol{\mu}}\hat{\mathbf{v}}\_{\boldsymbol{\mu}}^{\rm KJ}(\boldsymbol{h}^{\rm IK} - \boldsymbol{z}^{\rm KJ}), \ \hat{\mathbf{v}}\_{\boldsymbol{\sigma}}^{\rm IK}(\boldsymbol{z}^{\rm IK}) = -\mathbf{T}\_{\boldsymbol{\sigma}}\hat{\mathbf{v}}\_{\boldsymbol{\sigma}}^{\rm KJ}(\boldsymbol{h}^{\rm IK} - \boldsymbol{z}^{\rm KJ})\tag{46}
$$

where *JK h* ( *KJ h* ) denotes the thickness of layer *JK* (*KJ*). Substitution of Eqs. (35) and (36) into Eq. (46), one obtains, by noticing *JK KJ* **Λ Λ** , *JK KJ <sup>v</sup>* **<sup>Φ</sup> <sup>T</sup> <sup>Φ</sup>** , <sup>1</sup> *v v* **T T** , *JK KJ a d n n* and *JK KJ d a n n* , the local phase relation of a typical layer *j* (i.e. *JK* or *KJ*)

$$
\begin{bmatrix} \mathbf{a}^{\mathrm{IK}} \\ \mathbf{a}^{\mathrm{K}\mathbf{j}} \end{bmatrix} = \begin{bmatrix} \exp(-\mathbf{A}\_{-}^{\mathrm{IK}}h^{\mathrm{IK}}) & \mathbf{0} \\ \mathbf{0} & \exp(\mathbf{A}\_{+}^{\mathrm{IK}}h^{\mathrm{IK}}) \end{bmatrix} \begin{bmatrix} \mathbf{d}^{\mathrm{K}j} \\ \mathbf{d}^{\mathrm{IK}} \end{bmatrix} = \begin{bmatrix} \mathbf{P}^{\mathrm{IK}} & \mathbf{0} \\ \mathbf{0} & \mathbf{P}^{\mathrm{K}j} \end{bmatrix} \begin{bmatrix} \mathbf{0} & \mathbf{I}\_{a}^{\mathrm{IK}} \\ \mathbf{I}\_{d}^{\mathrm{IK}} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{d}^{\mathrm{IK}} \\ \mathbf{d}^{\mathrm{K}j} \end{bmatrix} \tag{47}
$$

where exp( ) *JK JK JK <sup>h</sup>* **<sup>P</sup> Λ** and exp( ) *KJ JK JK <sup>h</sup>* **<sup>P</sup> <sup>Λ</sup>** are respectively the *JK JK a a n n* and *JK JK d d n n* diagonal local phase matrices, and *JK <sup>a</sup>* **<sup>I</sup>** and *JK d***I** are identity matrices of order *JK a n* and *JK <sup>d</sup> n* , respectively.

Grouping together the local phase relations for all layers from up to down, one obtains the global phase relation

$$\mathbf{a} = \mathbf{P} \mathbf{U} \mathbf{d} \tag{48}$$

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 289

Material parameters

 2330 11 22 33 

 3512 11 22 9 ,

Dielectric constant ( <sup>12</sup> 8.854 10 F/m)

1.6 —

3.9 —

11.8 —

33 11 Piezoelectric constants (C/m2)

15 24 *e e* 0.48, 31 32 *e e* 0.58, <sup>33</sup> *e* 1.55

for the

 / 2

Mass density (kg/m3)

2700

2200

convenience of engineering application, of various waves in the multi-layered BAR are calculated by the formulation presented in Section 3.4 as the wavenumbers *<sup>x</sup> k* and *<sup>y</sup> k* are set to be zero. In order to show the influence of the number of unit cells in the Bragg Cell on the wave characteristics, Bragg Cells with 2 and 5 unit cells are respectively considered. Moreover, for sake of exploring the effects of electrodes, Bragg Cell and substrate on the wave characteristics in the propagation medium, the resonant frequencies of the 3.0μm AlN film, 3.0μm AlN film with top and bottom electrodes, and the bulk acoustic resonator without substrate and with 5 unit cells in the Bragg Cell are also calculated. The obtained first fifteen resonant frequencies of these multi-layered structures are listed and compared in

In this section, the above proposed formulation of MRRM for analyzing the propagation characteristics of various waves in the integrated acoustic wave devices are validated by a bulk acoustic resonator (BAR) consisting of 0.3μm Al film as the top electrode, 3.0μm AlN film as the propagation medium, 0.3μm Al film as the bottom electrode, alternate 0.81μm SiO2 and 1.76μm AlN layers as the Bragg Cell, 0.81μm SiO2 layer as the support medium and 42.6μm Si layer as the substrate. The material parameters of the exemplified BAR used

**3.5. Numerical examples** 

in the calculation are given in Table 1.

Si

AlN

Elastic constants (GPa)

11 22 33 *ccc* 164.8 12 13 23 *ccc* 63.5 , 44 55 *c c* 79 , <sup>66</sup> *c* 50.65

11 22 *c c* 345 , <sup>33</sup> *c* 395 , <sup>12</sup> *c* 125 , 13 23 *c c* 120 , 44 55 *c c* 118 , <sup>66</sup> *c* 110

**Table 1.** Material properties of the exemplified bulk acoustic resonator

The resonant frequencies, represented by the engineering frequency *f*

Al *E* 69 , 26 *G*

SiO2 *E* 70 , 29.915 *G*

Type of

Isotropic elastic material

Transversely isotropic elastic material

Transversely isotropic piezoelectric material

Table 2.

material Material

where **P** and **U** are respectively the blocked diagonal global phase matrix and global permutation matrix composed of

$$\mathbf{P} = \mathbf{C}^{12}, \mathbf{P}^{21}, \mathbf{P}^{23}, \dots, \mathbf{P}^{ll}, \mathbf{P}^{JK}, \dots, \mathbf{P}^{N(N+1)}, \mathbf{P}^{(N+1)N} > \tag{49}$$

$$\mathbf{U} = \mathbf{U}^{12}, \mathbf{U}^{23}, \cdots, \mathbf{U}^{IK}, \cdots, \mathbf{U}^{N(N+1)} > \text{, } \quad \mathbf{U}\_{n\_v \times n\_v}^{\text{IK}} = \begin{bmatrix} \mathbf{0} & \mathbf{I}\_d^{\text{IK}} \\ \mathbf{I}\_d^{\text{IK}} & \mathbf{0} \end{bmatrix} \tag{50}$$

#### *3.4.5. System equation and dispersion equation*

The global scattering relation in Eq. (43) and the global phase relation in Eq. (48) both contain *<sup>v</sup> n N* equations for the *<sup>v</sup> n N* unknown arriving wave amplitudes (in **a** ) and *<sup>v</sup> n N* unknown departing wave amplitudes (in **d** ). Thus the wave vectors **a** and **d** can be determined accordingly. Substitution of Eq. (48) into Eq. (43) gives the system equation

$$\mathbf{P} \left( \mathbf{APU} + \mathbf{D} \right) \mathbf{d} = \mathbf{Rd} = \mathbf{s}\_0 \tag{51}$$

where **R APU D** is the system matrix.

If there is no excitation ( <sup>0</sup> **s 0** ), i.e. the free wave propagation problem is considered, the vanishing of the system matrix determinant yields the following dispersion equation

$$\left| \mathbf{R}(k\_{\chi'}k\_y;\alpha) \right| = \mathbf{0} \tag{52}$$

which may be solved numerically by a proper root searching technique (Guo, 2008). Thus, the complete propagation characteristics of various waves can be obtained. In particular, the resonant frequency of the multi-layered structures can be obtained as =0 *x y k k* .

It should be noted that the above proposed formulation of MRRM (Guo & Chen, 2008a, 2008b; Guo, 2008; Guo et al., 2009) excludes any exponentially growing function and matrix inversion, therefore possesses unconditionally numerical stability and enables inclusion of surface and interface wave modes.

## **3.5. Numerical examples**

288 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

*d d n n* diagonal local phase matrices, and *JK*

*3.4.5. System equation and dispersion equation* 

where **R APU D** is the system matrix.

surface and interface wave modes.

*JK JK*

*<sup>d</sup> n* , respectively.

permutation matrix composed of

global phase relation

and *JK*

where exp( ) *JK JK JK <sup>h</sup>* **<sup>P</sup> Λ** and exp( ) *KJ JK JK <sup>h</sup>* **<sup>P</sup> <sup>Λ</sup>** are respectively the *JK JK*

Grouping together the local phase relations for all layers from up to down, one obtains the

where **P** and **U** are respectively the blocked diagonal global phase matrix and global

12 23 ( 1) , ,, ,, ,

*<sup>a</sup>* **<sup>I</sup>** and *JK*

*a a n n* and

*a n*

*d***I** are identity matrices of order *JK*

**a PUd** (48)

*JK*

(52)

12 21 23 ( 1) ( 1) , , ,, , ,, , *JI JK N N N N* **P P PP PP P P** (49)

*JK N N JK a*

 **0 I U UU U U U**

The global scattering relation in Eq. (43) and the global phase relation in Eq. (48) both contain *<sup>v</sup> n N* equations for the *<sup>v</sup> n N* unknown arriving wave amplitudes (in **a** ) and *<sup>v</sup> n N* unknown departing wave amplitudes (in **d** ). Thus the wave vectors **a** and **d** can be determined accordingly. Substitution of Eq. (48) into Eq. (43) gives the system equation

If there is no excitation ( <sup>0</sup> **s 0** ), i.e. the free wave propagation problem is considered, the

( , ;) *x y* **R 0** *k k* 

which may be solved numerically by a proper root searching technique (Guo, 2008). Thus, the complete propagation characteristics of various waves can be obtained. In particular, the

It should be noted that the above proposed formulation of MRRM (Guo & Chen, 2008a, 2008b; Guo, 2008; Guo et al., 2009) excludes any exponentially growing function and matrix inversion, therefore possesses unconditionally numerical stability and enables inclusion of

vanishing of the system matrix determinant yields the following dispersion equation

resonant frequency of the multi-layered structures can be obtained as =0 *x y k k* .

*v v*

*n n JK d*

**I 0** (50)

<sup>0</sup> ( ) **APU D d Rd s** (51)

In this section, the above proposed formulation of MRRM for analyzing the propagation characteristics of various waves in the integrated acoustic wave devices are validated by a bulk acoustic resonator (BAR) consisting of 0.3μm Al film as the top electrode, 3.0μm AlN film as the propagation medium, 0.3μm Al film as the bottom electrode, alternate 0.81μm SiO2 and 1.76μm AlN layers as the Bragg Cell, 0.81μm SiO2 layer as the support medium and 42.6μm Si layer as the substrate. The material parameters of the exemplified BAR used in the calculation are given in Table 1.


**Table 1.** Material properties of the exemplified bulk acoustic resonator

The resonant frequencies, represented by the engineering frequency *f* / 2 for the convenience of engineering application, of various waves in the multi-layered BAR are calculated by the formulation presented in Section 3.4 as the wavenumbers *<sup>x</sup> k* and *<sup>y</sup> k* are set to be zero. In order to show the influence of the number of unit cells in the Bragg Cell on the wave characteristics, Bragg Cells with 2 and 5 unit cells are respectively considered. Moreover, for sake of exploring the effects of electrodes, Bragg Cell and substrate on the wave characteristics in the propagation medium, the resonant frequencies of the 3.0μm AlN film, 3.0μm AlN film with top and bottom electrodes, and the bulk acoustic resonator without substrate and with 5 unit cells in the Bragg Cell are also calculated. The obtained first fifteen resonant frequencies of these multi-layered structures are listed and compared in Table 2.


Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 291

(1) In the SH-wave band structures of layered Bragg Cell, the phase constant spectra in passbands and the attenuation constant spectra in stop-bands occur alternately. The phase constant spectra of characteristic SH waves are formed from the dispersion curves of equivalent SH waves due to the zone folding effect and wave interference phenomenon. All the attenuation constant loops as *k* 0 and the second and upper attenuation constant loops as *k* 0 of characteristic SH waves are formed due to the separation of the dispersion curves of equivalent SH waves with respect to frequency during the forming of the phase spectra. The first attenuation constant loop as *k* 0 of characteristic SH wave is formed due to the cutoff property of SH waves in constituent layers. The contrasts of SH-wave characteristic impedances of the constituent layers, the characteristic time of the unit cell and the characteristic times of the constituent layers are three kinds of essential parameters determining the formation of the band structures. The contrasts of SH-wave characteristic impedances decide whether the stop-bands due to periodicity of the periodic layered media exist or not. If yes, it further decides the widths of the frequency bands. The characteristic time of the unit cell decides how many pass-bands/stop-bands exist in a specified frequency range. The characteristic times of the constituent layers mainly decides the mid-frequencies of the frequency bands. These rules can be used for the design of the layered Bragg Cell

(2) The proposed MRRM for integrated multi-layered acoustic wave devices is analytical based on distributed-parameter model, yields unified formulation, includes all wave modes and possesses unconditionally numerical stability. It therefore leads to high accurate results at small computational cost and is applicable to complex multilayered acoustic wave devices

(3) The integrated model considers nearly all the components in practical multi-layered acoustic wave devices, which definitely renders accurate wave propagation characteristics for guiding the proper design of and suppresses unfavorable spurious modes in the devices. Generally, the electrodes raise the resonant frequencies, while the Bragg Cell and the

In summary, the MRRM, the understanding of SH wave bands in the Bragg Cell and the integrated modeling of multi-layered acoustic wave devices with Bragg Cell in this chapter

will push forward the design of high-performed acoustic wave devices.

*Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of* 

*Education, and School of Civil Engineering and Mechanics, Lanzhou University,* 

*Department of Engineering Mechanics, Zhejiang University,P.R.China* 

according to SH-wave bands requirements.

substrate reduce the resonant frequencies.

**Author details** 

Yongqiang Guo

*P.R.China* 

Weiqiu Chen

while combined with a uniform computer program.

290 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

**Table 2.** Effects of components on the first fifteen frequencies of the exemplified BAR (GHz)

From Table 2, it is seen that all component layers in the multi-layered bulk acoustic wave device have obvious influence on the wave propagation characteristics, which validates the necessity to model the multi-layered acoustic wave devices by an integrated model with all components considered. The electrodes generally raise the resonant frequencies in the propagation medium except for the first mode. Adding unit cells of the Bragg Cell and the appending of substrate in the multilayered BAR will reduce the resonant frequencies and increase the number of wave modes in a given frequency range. These findings about the effects of electrodes, Bragg Cell and substrate on wave characteristics in the multilayered acoustic wave devices can be used in the design of these devices.

## **4. Conclusion**

The accurate analysis and design of layered Bragg Cell and of multi-layered acoustic wave devices with Bragg Cell are studied by the method of reverberation-ray matrix in this chapter. We obtain the analysis formulation, the features and the formation of SH-wave band structures in layered Bragg Cell and the design rules of layered Bragg Cell according to SH-wave band requirements. A unified formulation of MRRM is attained for the analysis of multi-layered acoustic wave devices modeled by integrated multi-layers consisting of working media, electrodes, Bragg Cell, support layer and substrate. The effects of other components on the resonant characteristics in the working media are gained. All findings are validated by numerical examples. The study in this chapter leads to the following conclusions:

(1) In the SH-wave band structures of layered Bragg Cell, the phase constant spectra in passbands and the attenuation constant spectra in stop-bands occur alternately. The phase constant spectra of characteristic SH waves are formed from the dispersion curves of equivalent SH waves due to the zone folding effect and wave interference phenomenon. All the attenuation constant loops as *k* 0 and the second and upper attenuation constant loops as *k* 0 of characteristic SH waves are formed due to the separation of the dispersion curves of equivalent SH waves with respect to frequency during the forming of the phase spectra. The first attenuation constant loop as *k* 0 of characteristic SH wave is formed due to the cutoff property of SH waves in constituent layers. The contrasts of SH-wave characteristic impedances of the constituent layers, the characteristic time of the unit cell and the characteristic times of the constituent layers are three kinds of essential parameters determining the formation of the band structures. The contrasts of SH-wave characteristic impedances decide whether the stop-bands due to periodicity of the periodic layered media exist or not. If yes, it further decides the widths of the frequency bands. The characteristic time of the unit cell decides how many pass-bands/stop-bands exist in a specified frequency range. The characteristic times of the constituent layers mainly decides the mid-frequencies of the frequency bands. These rules can be used for the design of the layered Bragg Cell according to SH-wave bands requirements.

(2) The proposed MRRM for integrated multi-layered acoustic wave devices is analytical based on distributed-parameter model, yields unified formulation, includes all wave modes and possesses unconditionally numerical stability. It therefore leads to high accurate results at small computational cost and is applicable to complex multilayered acoustic wave devices while combined with a uniform computer program.

(3) The integrated model considers nearly all the components in practical multi-layered acoustic wave devices, which definitely renders accurate wave propagation characteristics for guiding the proper design of and suppresses unfavorable spurious modes in the devices. Generally, the electrodes raise the resonant frequencies, while the Bragg Cell and the substrate reduce the resonant frequencies.

In summary, the MRRM, the understanding of SH wave bands in the Bragg Cell and the integrated modeling of multi-layered acoustic wave devices with Bragg Cell in this chapter will push forward the design of high-performed acoustic wave devices.

## **Author details**

290 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

The whole BAR with 2 unit cells in the Bragg Cell

1 0.9750 0.8380 0.0523 0.0441 0.1275 2 1.8217 2.2329 0.1038 0.0655 0.2517 3 1.9328 3.3316 0.1564 0.0885 0.3810 4 2.9014 4.8778 0.2096 0.1332 0.5130 5 3.6438 5.2997 0.2608 0.1755 0.6320 6 3.8643 5.7110 0.3106 0.2031 0.8749 7 4.8326 6.5112 0.3645 0.2206 1.0236 8 5.4657 8.9604 0.4771 0.2652 1.2409 9 5.7971 9.7544 0.5747 0.2668 1.3322 10 6.7639 10.5897 0.6334 0.3066 1.4596 11 7.2868 12.1454 0.6964 0.3350 1.7058 12 7.7296 13.2501 0.7610 0.3542 1.7985 13 8.6956 14.6379 0.8262 0.3984 1.9343 14 9.1095 15.4651 0.8727 0.4394 2.1422 15 9.6604 16.2959 0.8943 0.4679 2.2865

**Table 2.** Effects of components on the first fifteen frequencies of the exemplified BAR (GHz)

acoustic wave devices can be used in the design of these devices.

From Table 2, it is seen that all component layers in the multi-layered bulk acoustic wave device have obvious influence on the wave propagation characteristics, which validates the necessity to model the multi-layered acoustic wave devices by an integrated model with all components considered. The electrodes generally raise the resonant frequencies in the propagation medium except for the first mode. Adding unit cells of the Bragg Cell and the appending of substrate in the multilayered BAR will reduce the resonant frequencies and increase the number of wave modes in a given frequency range. These findings about the effects of electrodes, Bragg Cell and substrate on wave characteristics in the multilayered

The accurate analysis and design of layered Bragg Cell and of multi-layered acoustic wave devices with Bragg Cell are studied by the method of reverberation-ray matrix in this chapter. We obtain the analysis formulation, the features and the formation of SH-wave band structures in layered Bragg Cell and the design rules of layered Bragg Cell according to SH-wave band requirements. A unified formulation of MRRM is attained for the analysis of multi-layered acoustic wave devices modeled by integrated multi-layers consisting of working media, electrodes, Bragg Cell, support layer and substrate. The effects of other components on the resonant characteristics in the working media are gained. All findings are validated by

numerical examples. The study in this chapter leads to the following conclusions:

The whole BAR with 5 unit cells in the Bragg Cell

The whole BAR without substrate and with 5 unit cells in the Bragg Cell

3.0μm AlN film with electrodes

Order

3.0μm AlN film

**4. Conclusion** 

Yongqiang Guo *Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education, and School of Civil Engineering and Mechanics, Lanzhou University, P.R.China* 

Weiqiu Chen *Department of Engineering Mechanics, Zhejiang University,P.R.China* 

## **Acknowledgement**

This study was financially supported by the National Natural Science Foundation of China (Nos. 10902045 and 11090333), the Postdoctoral Science Foundation of China (Nos. 20090460155 and 201104019) and the Fundamental Research Funds for the Central Universities of China (Grant No. lzujbky-2011-9).

Precise Analysis and Design of Multi-Layered Acoustic Wave Devices with Bragg Cell 293

Guo, Y.Q. & Chen, W.Q. (2008b). Modeling of multilayered acoustic wave devices with the method of reverberation-ray matrix, *Symposium on Piezoelectricity, Acoustic Waves, and* 

Guo, Y.Q. (2008). *The Method of Reverberation-Ray Matrix and its Applications*, Doctorial

Guo, Y.Q., Chen, W.Q. & Zhang, Y.L. (2009). Guided wave propagation in multilayered piezoelectric structures, *Science in China*, *Series G*: *Physics*, *Mechanics and Astronomy*,

Hashimoto, K., Omori, T. & Yamaguchi, M. (2009). Characterization of surface acoustic wave propagation in multi-layered structures using extended FEM/SDA software, *IEEE* 

Kirsch, P., Assouar, M.B., Elmazria, O., Mortet, V., & Alnot, P. (2006). 5GHz surface acoustic wave devices based on aluminum nitride/diamond layered structure realized using

Lakin, K.M. (2005). Thin film resonator technology, *IEEE Transactions on Ultrasonics,* 

Lowe, M.J.S. (1995). Matrix techniques for modeling ultrasonic waves in multilayered media, *IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control*, 42: 525-542. Makkonen, T. (2005). *Numerical Simulations of Microacoustic Resonators and Filters*, Doctoral

Marechal, P., Haumesser, L., Tran-Huu-Hue, L.P., Holc, J., Kuscer, D., Lethiecq, M. & Feuillard, G. (2008). Modeling of a high frequency ultrasonic transducer using periodic

Mead, D.J. (1996). Wave propagation in continuous periodic structures: Research contributions from Southampton, 1964–1995. *Journal of Sound and Vibration*, 190(3): 495–

Nakanishi, H., Nakamura, H., Hamaoka, Y., Kamiguchi, H. & Iwasaki, Y. (2008). Small-sized SAW duplexers with wide duplex gap on a SiO2/Al/LiNbO3 structure by using novel Rayleigh-mode spurious suppression technique, *Proceedings of IEEE Ultrasonics* 

Naumenko, N.F. (2010). A universal technique for analysis of acoustic waves in periodic grating sandwiched between multi-layered structures and its application to different types of waves, *Proceedings of IEEE Ultrasonics Symposium*, *1673-1676*, San Diego, USA,

Pao, Y.H., Chen, W.Q. & Su, X.Y. (2007). The reverberation-ray matrix and transfer matrix

Pastureaud, T., Laude, V. & Ballandras, S. (2002). Stable scattering-matrix method for surface acoustic waves in piezoelectric multilayers, *Applied Physics Letters*, 80: 2544-2546. Royer, D. & Dieulesaint, E. (2000). *Elastic Waves in Solids I: Free and Guided Propagation*,

analyses of unidirectional wave motion, *Wave Motion*, 44: 419-438.

*Transactions on Ultrasonics, Ferroelectrics, and Frequency Control*, 56(11): 2559–2564. Hashimoto, K.Y. (2000). *Surface Acoustic Wave Devices in Telecommunications: Modeling and* 

dissertation. Zhejiang University, Hangzhou, China. (in Chinese)

electron beam lithography, *Applied Physics Letters*, 88: 223504.

Dissertation. Helsinki University of Technology, Espoo, Finland.

*Ferroelectrics, and Frequency Control*, 52(5): 707-716.

*Symposium*, 1588-1591, Beijing, China, Nov. 2008.

*Device Applications* (SPAWDA 2008), 105-110.

52(7): 1094-1104.

*Simulation*, Springer, Berlin.

structures, *Ultrasonics*, 48: 141–149.

524.

Oct. 2010.

Springer, Berlin.

## **5. References**


Guo, Y.Q. & Chen, W.Q. (2008b). Modeling of multilayered acoustic wave devices with the method of reverberation-ray matrix, *Symposium on Piezoelectricity, Acoustic Waves, and Device Applications* (SPAWDA 2008), 105-110.

292 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

Universities of China (Grant No. lzujbky-2011-9).

Dover Publications, New York.

*Frequency Control*, 55(2): 442–450.

Hill, New York.

York.

Publishers, New York.

ISBN: 978-953-307-111-4): 25-46.

*Acta Mechanica Solida Sinica*, 21: 500-506.

*Symposium, 250-253*, Beijing, China, Nov. 2008.

*of High Speed Electronics and Systems*, 10(3): 653-684.

integral formulation. *Journal of Applied Physics*, 96(12): 7731-7741.

*IEEE Ultrasonics Symposium, 1924-1927*, Beijing, China, Nov. 2008.

This study was financially supported by the National Natural Science Foundation of China (Nos. 10902045 and 11090333), the Postdoctoral Science Foundation of China (Nos. 20090460155 and 201104019) and the Fundamental Research Funds for the Central

Adler, E.L. (2000). Bulk and surface acoustic waves in anisotropic solids, *International Journal* 

Ballandras, S., Reinhardt, A., Laude, V., Soufyane, A., Camou, S., Daniau, W., Pastureaud, T., Steichen, W., Lardat, R., Solal, M. & Ventura, P. (2004) Simulations of surface acoustic wave devices built on stratified media using a mixed finite element/boundary

Benetti, M., Cannata, D., Di Pietrantonio, F., Verona, E., Almaviva, S., Prestopino, G., Verona, C. & Verona-Rinati, G. (2008). Surface acoustic wave devices on AlN/singlecrystal diamond for high frequency and high performances operation, *Proceedings of* 

Benetti, M., Cannatta, D., Di Pietrantonio, F. & Verona, E. (2005). Growth of AlN piezoelectric film on diamond for high-frequency surface acoustic wave devices, *IEEE* 

Brizoual, L.L., Sarry, F., Elmazria, O., Alnot, P., Ballandras, S. & Pastureaud, T. (2008). GHz frequency ZnO/Si SAW device, *IEEE Transactions on Ultrasonics, Ferroelectrics, and* 

Chung, C.-J. Chen, Y.-C., Cheng, C.-C., Wang, C.-M. & Kao, K.-S. (2008). Superior dual mode resonances for 1/4λ solidly mounted resonators, *Proceedings of IEEE Ultrasonics* 

Coddington, E. A., Levinson, N. (1955). *Theory of Ordinary Differential Equations*, McGraw-

Ding, H.J. & Chen, W.Q. (2001). *Three Dimensional Problems of Piezoelasticity*. Nova Science

Eringen, A.C. & Suhubi, E.S. (1975). *Elastodynamics II: Linear Theory*, Academic Press, New

Guo, Y.Q. & Chen, W.Q. (2010). Reverberation-ray matrix analysis of acoustic waves in multilayered anisotropic structures. *Acoustic Waves* (Edited by Don W. Dissanayake,

Guo, Y.Q. & Chen, W.Q. (2008a). On free wave propagation in anisotropic layered media,

*Transactions on Ultrasonics, Ferroelectrics, and Frequency Control*, 52: 1806–1811. Brillouin, L. (1953). *Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices*,

**Acknowledgement** 

**5. References** 

	- Shen, M.R. & Cao, W.W. (2000). Acoustic bandgap formation in a periodic structure with multilayer unit cells. *Journal of Physics D: Applied Physics*, 33: 1150–1154.

**Section 4** 

**Design and Fabrication of Microdevices** 


**Design and Fabrication of Microdevices** 

294 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

*Symposium*, 181-184, San Diego, USA, Oct. 2010.

*Physics*, 41: 77-103.

2016-2023.

4049-4051.

49: 142-149.

55(3): 704–716.

*Structures*, 39: 5447-5463.

*Solids and Structures*, 39: 5173-5184

Shen, M.R. & Cao, W.W. (2000). Acoustic bandgap formation in a periodic structure with

Stroh, A.N. (1962). Steady state problems in anisotropic elasticity, *Journal of Mathematics and* 

Su, X.Y., Tian, J.Y. & Pao, Y.H. (2002). Application of the reverberation-ray matrix to the propagation of elastic waves in a layered solid, *International Journal of Solids and* 

Tajic, A., Volatier, A., Aigner, R., Solal, M. (2010). Simulation of solidly mounted BAW resonators using FEM combined with BEM and/or PML, *Proceedings of IEEE Ultrasonics* 

Tan, E.L. (2007). Matrix Algorithms for modeling acoustic waves in piezoelectric multilayers, *IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control*, 54:

Tarn, J. Q. (2002a). A state space formalism for anisotropic elasticity. Part I: Rectilinear

Tarn, J. Q. (2002b). A state space formalism for piezothermoelasticity, *International Journal of* 

Wang, G., Yu, D.L., Wen, J.H., Liu, Y.Z., Wen, X.S. (2004). One-dimensional phononic

Wang, L. & Rokhlin, S.I. (2002). Recursive asymptotic stiffness matrix method for analysis of surface acoustic wave devices on layered piezoelectric media, *Applied Physics Letters*, 81:

Wu, T.T. & Chen, Y.Y. (2002) Exact analysis of dispersive SAW devices on ZnO/diamond/Silayered structures, *IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control*,

Yoon, G., & Park, J.-D. (2000). Fabrication of ZnO-based film bulk acoustic resonator devices

Zhang, V.Y., Dubus, B., Lefebvre, J.E. & Gryba, T. (2008). Modeling of bulk acoustic wave devices built on piezoelectric stack structures: Impedance matrix analysis and network representation, *IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control*,

Zhang, V.Y., Lefebvre, J.E. & Gryba, T. (2006). Resonant transmission in stop bands of

using W/SiO2 multilayer reflector, *IEEE Electronics Letters*, 36(16): 1435-1437.

acoustic waves in periodic structures, *Ultrasonics*, 44(1): 899–904.

anisotropy, *International Journal of Solids and Structures*, 39: 5143-5155

crystals with locally resonant structures. *Physics Letters A*, 327: 512–521.

multilayer unit cells. *Journal of Physics D: Applied Physics*, 33: 1150–1154.

**Chapter 13** 

© 2013 Baron et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2013 Baron et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**High-Overtone Bulk Acoustic Resonator** 

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56175

**1. Introduction** 

substrates [4].

T. Baron, E. Lebrasseur, F. Bassignot, G. Martin, V. Pétrini and S. Ballandras

Piezoelectricity has been used for the development of numerous time&frequency passive devices [1]. Among all these, radio-frequency devices based on surface acoustic waves (SAW) or bulk acoustic waves (BAW) have received a very large interest for bandpass filter and frequency source applications. Billions of these components are spread each year around the world due to their specific functionalities and the maturity of their related technologies [2]. The demand for highly coupled high quality acoustic wave devices has generated a strong innovative activity, yielding the investigation of new device structures. A lot of work has been achieved exploiting thin piezoelectric films for the excitation and detection of BAW to develop low loss RF filters [3]. However, problems still exist for selecting the layer orientation to favor specific mode polarization and select propagation characteristics (velocity, coupling, temperature sensitivity, *etc*.). Moreover, for given applications, deposited films reveal incapable to reach the characteristics of monolithic

For practical implementation, BAW is applied for standard low frequency (5 to 10MHz) shear wave resonators on Quartz for instance. SAW, Film Bulk Acoustic Resonator (FBAR) and High overtone Bulk Acoustic Resonator (HBAR) devices are applied for standard radiofrequency ranges and more particularly in S band (2 to 4GHz). HBAR have been particularly developed along different approaches to take advantage of their extremely high quality factor and very compact structure. Until now, many investigations have been carried out using piezoelectric thin films (Aluminum Nitride – AlN, Zinc Oxide – ZnO) atop thick wafers of silicon or sapphire [5] but recent developments showed the interest of thinned single-crystal-based structure in that purpose [6]. Although marginal, their application has been mainly focused on filters and frequency stabilization (oscillator) purposes [7], but the demonstration of their effective implementation for sensor applications has been achieved recently [8]. These devices maximize the *Q* factor that can be obtained at room temperature using elastic waves, yielding quality factor times Frequency products (*Q.f*) close or slightly

## **High-Overtone Bulk Acoustic Resonator**

T. Baron, E. Lebrasseur, F. Bassignot, G. Martin, V. Pétrini and S. Ballandras

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56175

## **1. Introduction**

Piezoelectricity has been used for the development of numerous time&frequency passive devices [1]. Among all these, radio-frequency devices based on surface acoustic waves (SAW) or bulk acoustic waves (BAW) have received a very large interest for bandpass filter and frequency source applications. Billions of these components are spread each year around the world due to their specific functionalities and the maturity of their related technologies [2]. The demand for highly coupled high quality acoustic wave devices has generated a strong innovative activity, yielding the investigation of new device structures. A lot of work has been achieved exploiting thin piezoelectric films for the excitation and detection of BAW to develop low loss RF filters [3]. However, problems still exist for selecting the layer orientation to favor specific mode polarization and select propagation characteristics (velocity, coupling, temperature sensitivity, *etc*.). Moreover, for given applications, deposited films reveal incapable to reach the characteristics of monolithic substrates [4].

For practical implementation, BAW is applied for standard low frequency (5 to 10MHz) shear wave resonators on Quartz for instance. SAW, Film Bulk Acoustic Resonator (FBAR) and High overtone Bulk Acoustic Resonator (HBAR) devices are applied for standard radiofrequency ranges and more particularly in S band (2 to 4GHz). HBAR have been particularly developed along different approaches to take advantage of their extremely high quality factor and very compact structure. Until now, many investigations have been carried out using piezoelectric thin films (Aluminum Nitride – AlN, Zinc Oxide – ZnO) atop thick wafers of silicon or sapphire [5] but recent developments showed the interest of thinned single-crystal-based structure in that purpose [6]. Although marginal, their application has been mainly focused on filters and frequency stabilization (oscillator) purposes [7], but the demonstration of their effective implementation for sensor applications has been achieved recently [8]. These devices maximize the *Q* factor that can be obtained at room temperature using elastic waves, yielding quality factor times Frequency products (*Q.f*) close or slightly

above 1014, *i.e.* effective *Q* factors of about 10,000 at 1GHz in theory (practically, *Q* factors in excess of 50,000 between 1.5 and 2GHz were experimentally achieved [9])

High-Overtone Bulk Acoustic Resonator 299

dominated by the acoustic property of the substrate. When the thickness of the substrate decreases, the device tends to behave as a FBAR (corresponding to *ts*=0 in fig.1). Depending on both the material and the cut orientations of piezoelectric transducer, pure longitudinal

or pure shear waves or combinations of these basic polarizations can be excited.

**Figure 2.** Schematic representation of the typical electrical response of HBARs.

modes.

Single port resonator structures can be easily achieved using HBARs, the use of two series devices being generally adopted to avoid etching the piezoelectric layer to reach the back electrode. Despite this favorable aspect, the exclusive use of single-port resonators limits HBAR applicability fields. Therefore, the possibility to fabricate four-port devices has been considered and experimentally tested (as shown in section 3.2). The leading idea consisted in transversely (or laterally) coupling acoustic waves between two adjacent resonators. The principle of such devices was inspired from the so-called monolithic filters based on coupled bulk waves in single crystals [10]. This is achieved by setting two resonators very close to one another. The gap between these resonators must be narrow enough to allow for the evanescent waves between the resonator electrodes to overlap and hence to yield mode coupling conditions. This system exhibits two eigenmodes with slightly different eigenfrequencies: a symmetric mode in which the coupled resonators vibrate in phase and an anti-symmetric mode in which they vibrate in phase opposition, as shown in figure 3. The gap between the two electrodes controls the spectral distance between the two coupled

HBAR-based sensors exploit two principal features yielding notable differences with other sensing solutions. The first one is related to the anisotropy of piezoelectric crystals on which these devices are built, which allows one for selecting crystal cut angles to optimize their physical characteristics. It is subsequently possible to choose cut angles to favor or minimize the parametric sensitivities of the considered wave propagation. The second remarkable feature of these devices concerns the use of piezoelectric excitation/detection of acoustoelectric waves which allows for wireless interrogation in radio-frequency ranges such as ISM bands centered at 434MHz, 868MHz, 915MHz or even 2.45GHz.

This chapter presents HBAR principles and related applications. Specific acoustic and electrical behaviors of HBAR are discussed and the different ways devoted to the manufacture of these devices also are presented. Applications of HBAR such as oscillator stabilization, intrinsically temperature-compensated devices and sensors are finally reported. Further developments required to promote the industrial exploitation of HBAR are discussed to conclude this article.

## **2. HBAR principles**

HBARs are constituted by a thin piezoelectric transducer above a high-quality acoustic substrate, as shown in figure 1. The piezoelectric transducer generates acoustic waves in the whole material stack along its effective electromechanical strength. Stationary waves are established between top and bottom free surfaces according to normal stress-free boundary conditions. As no electrical boundary condition arises at this surface, all the possible harmonics of the fundamental mode can exist. However, only the even modes of the transducer are excited as the only ones meeting the electrical boundary conditions applied to the transducer.

**Figure 1.** Schematic of HBAR

The electrical response of a HBAR can thus be interpreted as the modulation of the transducer resonance by the whole-stack bulk modes, presenting a dense spectrum of discrete modes localized around the resonance frequencies of the only piezoelectric transducer, as shown in figure 2. Since the substrate thickness is much larger than that of the piezoelectric film, most energy is stored in the substrate, and thus, the quality (*Q*) factor is dominated by the acoustic property of the substrate. When the thickness of the substrate decreases, the device tends to behave as a FBAR (corresponding to *ts*=0 in fig.1). Depending on both the material and the cut orientations of piezoelectric transducer, pure longitudinal or pure shear waves or combinations of these basic polarizations can be excited.

298 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

ISM bands centered at 434MHz, 868MHz, 915MHz or even 2.45GHz.

are discussed to conclude this article.

**2. HBAR principles** 

to the transducer.

**Figure 1.** Schematic of HBAR

excess of 50,000 between 1.5 and 2GHz were experimentally achieved [9])

above 1014, *i.e.* effective *Q* factors of about 10,000 at 1GHz in theory (practically, *Q* factors in

HBAR-based sensors exploit two principal features yielding notable differences with other sensing solutions. The first one is related to the anisotropy of piezoelectric crystals on which these devices are built, which allows one for selecting crystal cut angles to optimize their physical characteristics. It is subsequently possible to choose cut angles to favor or minimize the parametric sensitivities of the considered wave propagation. The second remarkable feature of these devices concerns the use of piezoelectric excitation/detection of acoustoelectric waves which allows for wireless interrogation in radio-frequency ranges such as

This chapter presents HBAR principles and related applications. Specific acoustic and electrical behaviors of HBAR are discussed and the different ways devoted to the manufacture of these devices also are presented. Applications of HBAR such as oscillator stabilization, intrinsically temperature-compensated devices and sensors are finally reported. Further developments required to promote the industrial exploitation of HBAR

HBARs are constituted by a thin piezoelectric transducer above a high-quality acoustic substrate, as shown in figure 1. The piezoelectric transducer generates acoustic waves in the whole material stack along its effective electromechanical strength. Stationary waves are established between top and bottom free surfaces according to normal stress-free boundary conditions. As no electrical boundary condition arises at this surface, all the possible harmonics of the fundamental mode can exist. However, only the even modes of the transducer are excited as the only ones meeting the electrical boundary conditions applied

The electrical response of a HBAR can thus be interpreted as the modulation of the transducer resonance by the whole-stack bulk modes, presenting a dense spectrum of discrete modes localized around the resonance frequencies of the only piezoelectric transducer, as shown in figure 2. Since the substrate thickness is much larger than that of the piezoelectric film, most energy is stored in the substrate, and thus, the quality (*Q*) factor is

**Figure 2.** Schematic representation of the typical electrical response of HBARs.

Single port resonator structures can be easily achieved using HBARs, the use of two series devices being generally adopted to avoid etching the piezoelectric layer to reach the back electrode. Despite this favorable aspect, the exclusive use of single-port resonators limits HBAR applicability fields. Therefore, the possibility to fabricate four-port devices has been considered and experimentally tested (as shown in section 3.2). The leading idea consisted in transversely (or laterally) coupling acoustic waves between two adjacent resonators. The principle of such devices was inspired from the so-called monolithic filters based on coupled bulk waves in single crystals [10]. This is achieved by setting two resonators very close to one another. The gap between these resonators must be narrow enough to allow for the evanescent waves between the resonator electrodes to overlap and hence to yield mode coupling conditions. This system exhibits two eigenmodes with slightly different eigenfrequencies: a symmetric mode in which the coupled resonators vibrate in phase and an anti-symmetric mode in which they vibrate in phase opposition, as shown in figure 3. The gap between the two electrodes controls the spectral distance between the two coupled modes.

High-Overtone Bulk Acoustic Resonator 301

off between the mode density and the stack thickness therefore is mandatory to optimize the HBAR response. Increasing the number of modes experimentally tends to provide higher quality coefficients for modes close to the transducer one but also reduces the corresponding coupling and significantly impact the device spectral density, yielding more difficulty to

**Figure 4.** Impact of acoustic substrate. The reflection parameter-S11 with respect to a 50 load is measured for different substrate thicknesses. A material stack consisting of an acoustic substrate of LiNbO3 (YX*l*)/163°, an aluminum electrode of 10nm thick, a 10µm thin piezoelectric layer of LiNbO3 (YX*l*)/163° and a 10nm thick bottom aluminum electrode is considered here for a theoretical description of the HBAR characteristics. Electrode thickness are chosen extremely thin to neglect their acoustic influence. Acoustic and dielectric losses are only consider in LiNbO3 layers for scaling the maximum achievable quality factors. For all computations, an active electrode surface of 100x100µm² has been

For a given stack, the coupling coefficient of each group of overtones (these groups being defined by fundamental and even overtones of the transducer alone) depends on the material coupling coefficient of the transducer and on the order of the considered group. Indeed, the third order group presents a coupling coefficient divided by 9 compared to the fundamental group (one third of the fundamental mode coupling at excitation times one third at detection), the fifth is divided by 25, and so on. LiNbO3 presents material coupling coefficient noticeably higher (3 to 7 times larger) than other material generally used for HBAR fabrication, such as AlN, ZnO. As a consequence, even transducer overtone groups can be effectively used with such a material and more especially when exciting pure shear waves as exposed further. Figure 5 shows the electrical response of a single-port HBAR built with (YX*l*)/163° LiNbO3 piezoelectric layer and substrate. Only shear waves are excited and all even group can be visible from the fundamental to the 11th harmonic of the layer alone

Each overtone in a given group presents a specific coupling coefficient *ks²* and a specific quality factor *Q*. Central overtones present the highest coupling coefficients within a group, but not always the highest quality factors. Indeed, the substrate (Sapphire for instance) is usually chosen with acoustic quality better than the transducer material (Aln, ZnO) as it is expected to act as the effective resonant cavity, whereas the transducer material is selected for its piezoelectric strength. As explained above, the energy ratio within the transducer and

exploit well defined resonance.

considered for normative purposes.

near 2GHz.

**Figure 3.** Principle scheme of the laterally-coupled-mode HBAR filter (a) symmetrical mode (b) antisymmetrical mode

## **3. HBAR devices**

## **3.1. Electrical and acoustic behavior**

## *3.1.1. Single-port resonator*

As explained above, the electrical response spectrum of such HBAR presents a large number of overtones. A large band representation allows for the observation of several envelopes themselves composed of several overtones. The central frequencies of these envelopes correspond to fundamental and even overtone resonances of the only transducer and therefore are mainly controlled by the transducer thickness.

Figure 4 shows the S11 response for the considered structure for different substrate thicknesses, illustrating the impact of this parameter on the overtone characteristics. The highest electromechanically-coupled overtone corresponds to the mode matching at best maximum energy location within the transducer, whereas the other overtones do exhibit smaller coupling factor proportionally to their spectral distance with the central overtone. The case of FBAR (*ts*=0 µm in figure 1) is reported on this graph to effectively localize the central frequency of resonance and anti-resonance of the only transducer. In presence of a substrate, mode coupling between the two layers is made possible and several overtones appear for substrate thickness larger than the transducer one, as illustrated in figure 4. As suggested previously, the spectral distance between two overtones is mainly due to the substrate properties (velocity and thickness) when the later exhibits a thickness much larger than the other layers of the whole stack. Figure 4 also shows the evolution of the electromechanical coupling coefficient (generally noted *ks²* for radio-frequency acoustoelectric devices) when increasing the substrate thickness. The reduction of *ks²* when increasing the substrate thickness is directly related to the energy ratio within the transducer and in the whole HBAR stack. Increasing the substrate thickness yields more energy in the whole HBAR structure and less energy within the transducer. Another interpretation consists in considering that the coupling of the transducer alone is spread on all the modes of the structure near the transducer resonance. Increasing the number of modes yields a reduction of the electromechanical of each mode coupled to the transducer mode. A tradeoff between the mode density and the stack thickness therefore is mandatory to optimize the HBAR response. Increasing the number of modes experimentally tends to provide higher quality coefficients for modes close to the transducer one but also reduces the corresponding coupling and significantly impact the device spectral density, yielding more difficulty to exploit well defined resonance.

300 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

symmetrical mode

**3. HBAR devices** 

*3.1.1. Single-port resonator* 

**3.1. Electrical and acoustic behavior** 

therefore are mainly controlled by the transducer thickness.

**Figure 3.** Principle scheme of the laterally-coupled-mode HBAR filter (a) symmetrical mode (b) anti-

As explained above, the electrical response spectrum of such HBAR presents a large number of overtones. A large band representation allows for the observation of several envelopes themselves composed of several overtones. The central frequencies of these envelopes correspond to fundamental and even overtone resonances of the only transducer and

Figure 4 shows the S11 response for the considered structure for different substrate thicknesses, illustrating the impact of this parameter on the overtone characteristics. The highest electromechanically-coupled overtone corresponds to the mode matching at best maximum energy location within the transducer, whereas the other overtones do exhibit smaller coupling factor proportionally to their spectral distance with the central overtone. The case of FBAR (*ts*=0 µm in figure 1) is reported on this graph to effectively localize the central frequency of resonance and anti-resonance of the only transducer. In presence of a substrate, mode coupling between the two layers is made possible and several overtones appear for substrate thickness larger than the transducer one, as illustrated in figure 4. As suggested previously, the spectral distance between two overtones is mainly due to the substrate properties (velocity and thickness) when the later exhibits a thickness much larger than the other layers of the whole stack. Figure 4 also shows the evolution of the electromechanical coupling coefficient (generally noted *ks²* for radio-frequency acoustoelectric devices) when increasing the substrate thickness. The reduction of *ks²* when increasing the substrate thickness is directly related to the energy ratio within the transducer and in the whole HBAR stack. Increasing the substrate thickness yields more energy in the whole HBAR structure and less energy within the transducer. Another interpretation consists in considering that the coupling of the transducer alone is spread on all the modes of the structure near the transducer resonance. Increasing the number of modes yields a reduction of the electromechanical of each mode coupled to the transducer mode. A trade-

**Figure 4.** Impact of acoustic substrate. The reflection parameter-S11 with respect to a 50 load is measured for different substrate thicknesses. A material stack consisting of an acoustic substrate of LiNbO3 (YX*l*)/163°, an aluminum electrode of 10nm thick, a 10µm thin piezoelectric layer of LiNbO3 (YX*l*)/163° and a 10nm thick bottom aluminum electrode is considered here for a theoretical description of the HBAR characteristics. Electrode thickness are chosen extremely thin to neglect their acoustic influence. Acoustic and dielectric losses are only consider in LiNbO3 layers for scaling the maximum achievable quality factors. For all computations, an active electrode surface of 100x100µm² has been considered for normative purposes.

For a given stack, the coupling coefficient of each group of overtones (these groups being defined by fundamental and even overtones of the transducer alone) depends on the material coupling coefficient of the transducer and on the order of the considered group. Indeed, the third order group presents a coupling coefficient divided by 9 compared to the fundamental group (one third of the fundamental mode coupling at excitation times one third at detection), the fifth is divided by 25, and so on. LiNbO3 presents material coupling coefficient noticeably higher (3 to 7 times larger) than other material generally used for HBAR fabrication, such as AlN, ZnO. As a consequence, even transducer overtone groups can be effectively used with such a material and more especially when exciting pure shear waves as exposed further. Figure 5 shows the electrical response of a single-port HBAR built with (YX*l*)/163° LiNbO3 piezoelectric layer and substrate. Only shear waves are excited and all even group can be visible from the fundamental to the 11th harmonic of the layer alone near 2GHz.

Each overtone in a given group presents a specific coupling coefficient *ks²* and a specific quality factor *Q*. Central overtones present the highest coupling coefficients within a group, but not always the highest quality factors. Indeed, the substrate (Sapphire for instance) is usually chosen with acoustic quality better than the transducer material (Aln, ZnO) as it is expected to act as the effective resonant cavity, whereas the transducer material is selected for its piezoelectric strength. As explained above, the energy ratio within the transducer and

the substrate is higher for the central overtones than for the overtones located at the edge of the group.

High-Overtone Bulk Acoustic Resonator 303

*3.1.2. Transversely-coupled HBAR* 

As explained above, the possibility to fabricate four-port devices has been considered and experimentally tested. Two HBAR resonators were fabricated on a LiNbO3 (34µm) / Au (300nm) / LiNbO3 (350µm) stack. Two 145x200µm2 surface aluminum electrodes, were patterned upon the stack and separated by a gap of 10µm. Figure 7 shows a typical coupledmode filter response for a device manufactured atop a LiNbO3/LiNbO3 structure. Rejection in excess of 20dB is demonstrated at 720MHz with a single filter cell. Insertion losses of about 15dB are emphasized and could be easily improved by impedance matching. The measured transfer function actually exhibits a double mode response, providing a first

evidence of the device operation according to the above assumptions.

**Figure 7.** Four-port laterally coupled HBAR devices 0.1% band 720MHz LiNbO3/LiNbO3 filter.

symmetrical and anti-symmetrical modes as experimentally observed.

**3.2. HBAR micro-fabrication** 

Furthermore, the following experiment was applied for definitely validating the lateral mode coupling. The admittance of the first resonator was measured for two different loading conditions applied to the second resonator (Figure 8). For open circuit conditions, a main peak corresponding to the first resonator contribution is observed (Figure 8, left) together with a weaker contribution near the main resonance. For short-circuit conditions, the admittance measured on the first resonator shows two almost-balanced resonance peaks, (Figure 8, right). This behavior is explained by the fact that no current crosses the second resonator when in open condition, yielding a small contribution of the anti-symmetrical mode (due to poor boundary condition matching) whereas loaded electrical condition allows for an effective excitation of the later mode, yielding almost balanced contributions of

Two main approaches can be implemented to manufacture HBAR devices. The first approach consists in physical or chemical deposition of thin piezoelectric layers (such as ZnO, PZT, AlN and so on) onto the chosen substrate. The first HBAR was manufactured along this approach [11]. The main advantage of this kind of HBAR is the capability of the

**Figure 5.** Single-port HBAR device built using LiNbO3/LiNbO3 (YX*l*)/163° cut.

In figure 6, a HBAR is constructed with LiNbO3 material for the transducer. As shown further, LiNbO3 presents better acoustic quality than Quartz, which is used for the HBAR substrate here to improve the device temperature stability (see section 4.2). In that example the overtone at 433.3MHz exhibits the best coupling coefficient *ks²* as well as the best quality factor *Q*, due to the acoustic quality of LiNbO3 compared to Quartz (see section 4.1).

According the above assumptions concerning material quality selection, the quality factor of overtones located at the edge of group is generally higher than the ones in the center of the group. Practically, it turns out that small-coupling overtones always exhibit better Q than the central overtones in a given group. One explanation of this objective result can be related to electrically generated losses (losses related to electrode resistivity and series resistance tends to increase with current).

**Figure 6.** 5th Envelope of single-port HBAR device constituted by LiNbO3 (YX*l*)/163°/Quartz.

#### *3.1.2. Transversely-coupled HBAR*

302 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

**Figure 5.** Single-port HBAR device built using LiNbO3/LiNbO3 (YX*l*)/163° cut.

tends to increase with current).

the group.

the substrate is higher for the central overtones than for the overtones located at the edge of

In figure 6, a HBAR is constructed with LiNbO3 material for the transducer. As shown further, LiNbO3 presents better acoustic quality than Quartz, which is used for the HBAR substrate here to improve the device temperature stability (see section 4.2). In that example the overtone at 433.3MHz exhibits the best coupling coefficient *ks²* as well as the best quality

According the above assumptions concerning material quality selection, the quality factor of overtones located at the edge of group is generally higher than the ones in the center of the group. Practically, it turns out that small-coupling overtones always exhibit better Q than the central overtones in a given group. One explanation of this objective result can be related to electrically generated losses (losses related to electrode resistivity and series resistance

factor *Q*, due to the acoustic quality of LiNbO3 compared to Quartz (see section 4.1).

**Figure 6.** 5th Envelope of single-port HBAR device constituted by LiNbO3 (YX*l*)/163°/Quartz.

As explained above, the possibility to fabricate four-port devices has been considered and experimentally tested. Two HBAR resonators were fabricated on a LiNbO3 (34µm) / Au (300nm) / LiNbO3 (350µm) stack. Two 145x200µm2 surface aluminum electrodes, were patterned upon the stack and separated by a gap of 10µm. Figure 7 shows a typical coupledmode filter response for a device manufactured atop a LiNbO3/LiNbO3 structure. Rejection in excess of 20dB is demonstrated at 720MHz with a single filter cell. Insertion losses of about 15dB are emphasized and could be easily improved by impedance matching. The measured transfer function actually exhibits a double mode response, providing a first evidence of the device operation according to the above assumptions.

**Figure 7.** Four-port laterally coupled HBAR devices 0.1% band 720MHz LiNbO3/LiNbO3 filter.

Furthermore, the following experiment was applied for definitely validating the lateral mode coupling. The admittance of the first resonator was measured for two different loading conditions applied to the second resonator (Figure 8). For open circuit conditions, a main peak corresponding to the first resonator contribution is observed (Figure 8, left) together with a weaker contribution near the main resonance. For short-circuit conditions, the admittance measured on the first resonator shows two almost-balanced resonance peaks, (Figure 8, right). This behavior is explained by the fact that no current crosses the second resonator when in open condition, yielding a small contribution of the anti-symmetrical mode (due to poor boundary condition matching) whereas loaded electrical condition allows for an effective excitation of the later mode, yielding almost balanced contributions of symmetrical and anti-symmetrical modes as experimentally observed.

#### **3.2. HBAR micro-fabrication**

Two main approaches can be implemented to manufacture HBAR devices. The first approach consists in physical or chemical deposition of thin piezoelectric layers (such as ZnO, PZT, AlN and so on) onto the chosen substrate. The first HBAR was manufactured along this approach [11]. The main advantage of this kind of HBAR is the capability of the

related techniques (sputtering, epitaxy, sol-gel spinning/firing, pulsed laser ablation, *etc*.) to deposit thin layers which allow for achieving device naturally operating at high frequency (for instance in the vicinity of the 2.45GHz ISM Band, or even more). This approach also did provide among the highest *Q* factor ever measured for an acoustic-based resonator at room temperature [3], with *Q.f* product values in excess of 1014 at parallel resonance (7.1013 at series resonance).

High-Overtone Bulk Acoustic Resonator 305

atop any material stack makes possible the use of specific crystal cut to select the polarization of the excited acoustic waves as well as its electromechanical coupling

The development of the so-called Silicon On Insulator (SOI(TM)) wafers has demonstrated the huge opportunities offered by the Smart Cut(TM) approach [12]. Moreover, its application for transferring single crystal Lithium Niobate thin layer into silicon proved to be effective for SAW device development [13]. As this technology requires a severe know-how and complex technological facilities and environment, an alternative fabrication technique based on metal diffusion at the interface between the materials to be bonded together [14] has been developed together with a lapping/polishing technique for HBAR manufacturing [15].

**Figure 9.** Process flow-chart for the fabrication of the HBAR based on bonding and lapping technology.

In this particular approach, contrary to the sputtering method, thermal process forbids to stack materials presenting notably differential thermal expansion coefficients. Smart Cut(TM) approach allows one to produce thin single crystal layers (such as LiNbO3 for instance, or even LiTaO3 [16]) with typical thickness below 1µm. Along this approach, embedded metal

The above-mentioned bonding and lapping technology has specifically been developed to allow for material stacking at room temperature, for exploiting any material of any crystal orientation. The process flow-chart reported in figure 9 allows one for a collective manufacturing of HBAR devices. This process is based on the mechanical diffusion of submicron gold layers, providing an effective acoustic liaison of the chosen material as well as the HBAR back electrode at once. As the bonding operation is achieved at room temperature, no significant thermal differential effects are observed and the resulting wafer can be handled and further processed provided thermal budget remains smaller than 100°C

Along the proposed approach, optical quality polished surfaces are preferred to favor the bonding of the wafers. A Chromium and Gold thin layer is first deposited by sputtering on

electrodes are fabricated using the Smart Cuttrade technology [16].

(as experimentally observed).

coefficient.

**Figure 8.** Admittances of one of the two resonators of the laterally-coupled structure as a function of the electrical conditions applied to the associated resonator (left) open circuit (right) 50Ω loading

Although this approach revealed efficient for operational device manufacturing, some drawbacks can be identified which limit the interest of the related resonators. Among these, one of the most problematic concerns the electromechanical coupling coefficient one can obtain particularly with AlN and ZnO, the most used thin piezoelectric layer for RF acoustic devices. As deposition techniques (principally reactive sputtering but also pulsed laser deposition (PLD)) generally allows for depositing well-controlled homogeneous C-oriented layers (i.e. with the C crystal axis oriented along the normal of the coated surface), the maximum accessible coupling remains much below 10%. Also the corresponding modes are purely longitudinal, with reduced degree-of-freedom for effectively controlled layer orientation suited for shear wave excitation/detection. Thin layers such as PZT can overcome this limitation but they generally exhibit notably high visco-elastic coefficients and significant dielectric loss which again limit their interest for high frequency (above 1GHz) applications. More generally, acoustic losses of most piezoelectric layers obtained by sputtering, sol-gel and techniques providing poly-crystalline materials always reveal larger than single-crystal ones. As explained before, the coupling coefficient of each high-overtone resonance depends on the number of overtones and on the intrinsic material electromechanical coupling coefficient. Poor material coupling coefficients prevent the use of overtones modulating the third (and therefore higher order) overtone of the piezoelectric transducer. Finally, compensating longitudinal modes thermal drift is particularly difficult as most of the high acoustic quality materials exhibit negative temperature coefficients of the corresponding phase velocity (as well as the transducer materials, ranging from -20 to -60ppm.K-1). These negative aspects pushed to seek for other manufacturing approaches.

The opportunity to use single crystal layers for acoustic transduction therefore appears as an alternative solution. Assuming the possibility for manufacturing thin single-crystal films atop any material stack makes possible the use of specific crystal cut to select the polarization of the excited acoustic waves as well as its electromechanical coupling coefficient.

304 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

series resonance).

for other manufacturing approaches.

related techniques (sputtering, epitaxy, sol-gel spinning/firing, pulsed laser ablation, *etc*.) to deposit thin layers which allow for achieving device naturally operating at high frequency (for instance in the vicinity of the 2.45GHz ISM Band, or even more). This approach also did provide among the highest *Q* factor ever measured for an acoustic-based resonator at room temperature [3], with *Q.f* product values in excess of 1014 at parallel resonance (7.1013 at

**Figure 8.** Admittances of one of the two resonators of the laterally-coupled structure as a function of the

Although this approach revealed efficient for operational device manufacturing, some drawbacks can be identified which limit the interest of the related resonators. Among these, one of the most problematic concerns the electromechanical coupling coefficient one can obtain particularly with AlN and ZnO, the most used thin piezoelectric layer for RF acoustic devices. As deposition techniques (principally reactive sputtering but also pulsed laser deposition (PLD)) generally allows for depositing well-controlled homogeneous C-oriented layers (i.e. with the C crystal axis oriented along the normal of the coated surface), the maximum accessible coupling remains much below 10%. Also the corresponding modes are purely longitudinal, with reduced degree-of-freedom for effectively controlled layer orientation suited for shear wave excitation/detection. Thin layers such as PZT can overcome this limitation but they generally exhibit notably high visco-elastic coefficients and significant dielectric loss which again limit their interest for high frequency (above 1GHz) applications. More generally, acoustic losses of most piezoelectric layers obtained by sputtering, sol-gel and techniques providing poly-crystalline materials always reveal larger than single-crystal ones. As explained before, the coupling coefficient of each high-overtone resonance depends on the number of overtones and on the intrinsic material electromechanical coupling coefficient. Poor material coupling coefficients prevent the use of overtones modulating the third (and therefore higher order) overtone of the piezoelectric transducer. Finally, compensating longitudinal modes thermal drift is particularly difficult as most of the high acoustic quality materials exhibit negative temperature coefficients of the corresponding phase velocity (as well as the transducer materials, ranging from -20 to -60ppm.K-1). These negative aspects pushed to seek

The opportunity to use single crystal layers for acoustic transduction therefore appears as an alternative solution. Assuming the possibility for manufacturing thin single-crystal films

electrical conditions applied to the associated resonator (left) open circuit (right) 50Ω loading

The development of the so-called Silicon On Insulator (SOI(TM)) wafers has demonstrated the huge opportunities offered by the Smart Cut(TM) approach [12]. Moreover, its application for transferring single crystal Lithium Niobate thin layer into silicon proved to be effective for SAW device development [13]. As this technology requires a severe know-how and complex technological facilities and environment, an alternative fabrication technique based on metal diffusion at the interface between the materials to be bonded together [14] has been developed together with a lapping/polishing technique for HBAR manufacturing [15].

**Figure 9.** Process flow-chart for the fabrication of the HBAR based on bonding and lapping technology.

In this particular approach, contrary to the sputtering method, thermal process forbids to stack materials presenting notably differential thermal expansion coefficients. Smart Cut(TM) approach allows one to produce thin single crystal layers (such as LiNbO3 for instance, or even LiTaO3 [16]) with typical thickness below 1µm. Along this approach, embedded metal electrodes are fabricated using the Smart Cuttrade technology [16].

The above-mentioned bonding and lapping technology has specifically been developed to allow for material stacking at room temperature, for exploiting any material of any crystal orientation. The process flow-chart reported in figure 9 allows one for a collective manufacturing of HBAR devices. This process is based on the mechanical diffusion of submicron gold layers, providing an effective acoustic liaison of the chosen material as well as the HBAR back electrode at once. As the bonding operation is achieved at room temperature, no significant thermal differential effects are observed and the resulting wafer can be handled and further processed provided thermal budget remains smaller than 100°C (as experimentally observed).

Along the proposed approach, optical quality polished surfaces are preferred to favor the bonding of the wafers. A Chromium and Gold thin layer is first deposited by sputtering on

both wafers to bond (LiNbO3 and Quartz in the example of figure 6 and 9). The LiNbO3 wafer is then bonded onto the substrate by mechanical compression of the 200nm thick gold layers into an EVG wafer bonding machine as shown in figure 10. During the bonding process, the material stack is kept at a temperature of 30°C and a pressure of 65N.cm−2 is applied on the whole contact surface. The bonding can be particularly controlled by adjusting the process duration and various parameters such as the applied pressure, the process temperature, the quality of the vacuum during the process, *etc*. In the reported development, the process temperature is kept near a value close to the final thermal conditions seen by the device in operation. Since substrate and piezoelectric materials have different thermal expansion coefficients, one must account for differential thermo-elastic stresses when bonding both wafers and minimize them as much as possible.

High-Overtone Bulk Acoustic Resonator 307

**Figure 11.** Ultrasonic characterization bench dedicated to non destructive control of the bonding

**Figure 12.** Characterization of a Silicon/LiNbO3 bonding – surfaces are bonded at 95%.

amplitude of the first detected signal contains the useful information.

 The control of the bonding can be made during the polishing steps without destruction; or the control can be done at the end of the process, indeed, the different layers

There is no constraint related to time resolution as in pulse-echo method, as the wafer

The analysis of the ultrasonic transmitted signals is very simple because only the

This method presents three major advantages:

**Figure 13.** SOMOS lapping/polishing machine.

obtained by sputtering do not disturb the measure.

thickness is not dramatically larger than the wavelength.

interface.

**Figure 10.** EVG wafer Bounder and illustration of Gold bonding process.

Once the bonding achieved, it is necessary to characterize the quality of the bonding. Due to the thickness of the wafers and the opacity of the stack (metal layers), optical measurements are poorly practicable. To avoid destructive controls of the material stack, ultrasonic techniques have been particularly considered here. The reliability of the bonding then is analyzed by ultrasonic transmission in a liquid environment. The bonded wafers are immersed in a water tank and the whole wafer stack surface is scanned. Figure 11 presents photography of the bench. Two focused transducers are used as acoustic emitter and receiver. They are manufactured by SONAXIS with a central frequency close to 50MHz, a 19mm active diameter and a 30mm focal length. The beam diameter at focal distance at -6dB is about 200µm.

Such a lateral resolution enables one to detect very small defects. The principle of the characterization method is based on the measurement of the received acoustic amplitude which depends on the variation of the acoustic impedance of the bonding area. If the bonding presents a defect at the interface between the two wafers, a dust or an air gap in most cases, the reflection coefficient of the incident wave is then nearly 1. The amplitude of the received wave is strongly reduced or even vanishes. Figure 12 shows a C-Scan of a Silicon/LiNbO3 wafer bonding characterization. The blue color corresponds to bonded surfaces, whereas yellow and green regions indicate bonding defects.

**Figure 11.** Ultrasonic characterization bench dedicated to non destructive control of the bonding interface.

**Figure 12.** Characterization of a Silicon/LiNbO3 bonding – surfaces are bonded at 95%.

This method presents three major advantages:

306 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

stresses when bonding both wafers and minimize them as much as possible.

**Figure 10.** EVG wafer Bounder and illustration of Gold bonding process.

surfaces, whereas yellow and green regions indicate bonding defects.

is about 200µm.

Once the bonding achieved, it is necessary to characterize the quality of the bonding. Due to the thickness of the wafers and the opacity of the stack (metal layers), optical measurements are poorly practicable. To avoid destructive controls of the material stack, ultrasonic techniques have been particularly considered here. The reliability of the bonding then is analyzed by ultrasonic transmission in a liquid environment. The bonded wafers are immersed in a water tank and the whole wafer stack surface is scanned. Figure 11 presents photography of the bench. Two focused transducers are used as acoustic emitter and receiver. They are manufactured by SONAXIS with a central frequency close to 50MHz, a 19mm active diameter and a 30mm focal length. The beam diameter at focal distance at -6dB

Such a lateral resolution enables one to detect very small defects. The principle of the characterization method is based on the measurement of the received acoustic amplitude which depends on the variation of the acoustic impedance of the bonding area. If the bonding presents a defect at the interface between the two wafers, a dust or an air gap in most cases, the reflection coefficient of the incident wave is then nearly 1. The amplitude of the received wave is strongly reduced or even vanishes. Figure 12 shows a C-Scan of a Silicon/LiNbO3 wafer bonding characterization. The blue color corresponds to bonded

both wafers to bond (LiNbO3 and Quartz in the example of figure 6 and 9). The LiNbO3 wafer is then bonded onto the substrate by mechanical compression of the 200nm thick gold layers into an EVG wafer bonding machine as shown in figure 10. During the bonding process, the material stack is kept at a temperature of 30°C and a pressure of 65N.cm−2 is applied on the whole contact surface. The bonding can be particularly controlled by adjusting the process duration and various parameters such as the applied pressure, the process temperature, the quality of the vacuum during the process, *etc*. In the reported development, the process temperature is kept near a value close to the final thermal conditions seen by the device in operation. Since substrate and piezoelectric materials have different thermal expansion coefficients, one must account for differential thermo-elastic


**Figure 13.** SOMOS lapping/polishing machine.

The piezoelectric wafer is subsequently thinned by lapping step to an overall thickness of 20µm. The lapping machine used in that purpose and shown in figure 13 is a SOMOS double side lapping/polishing machine based on a planetary motion of the wafers (up to 4" diameter) to promote abrasion homogeneity. An abrasive solution of silicon carbide is used here. The speed of the lapping is controlled by choosing the speed of rotation of the lapping machine stages, the load on the wafer, the rate of flow or the concentration of the abrasive. Once close to the expected thickness, the lapping process is followed by a micro-polishing step. This step uses similar equipments dedicated to polishing operation and hence using abrasive solution (colloidal silica) with smaller grain. This polishing step is applied until the average surface roughness *ra* remains larger than 3nm. Afterward, the wafer is considered ready for surface processing.

High-Overtone Bulk Acoustic Resonator 309

substrate exhibit *Q* factors of 53,000 at 1.5GHz using the Gold bonding technique [5] and *Q.f* product above 8.1013 with an 800nm thickness for the piezoelectric layer by Smart Cut approach [18]. Understanding losses phenomena helps to design high quality factor devices. Loss origins can be classified into two categories: material (intrinsic) and geometry (technology-related). Due to the architecture of HBAR, the quality factor of such devices depends on the crystalline losses and on the material isotropy, on the surfaces parallelism

**Figure 15.** Losses per wave length and the resonator's quality *Q* as a function of frequency (GHz) for

As explained before, the quality factor is directly link to the acoustic quality of the substrate. Some works have already been done to compare and improve materials to favor high

The polishing process providing damaged-free ultra-smooth surfaces is essential, as well as checking the substrate quality by X-ray topography. To take into account current industrial needs, using the technology of material crystal growth is crucial to obtain large wafers. In this context, LiNbO3, LiTaO3 Sapphire, and YAG are the preferred candidates as they do present effective acoustic quality (*i.e.* reduced visco-elastic and dielectric damping

The defect of parallelism between two surfaces of HBAR devices dramatically limits the quality factor [11]. Figure 16 shows the quality factor of HBAR modes on Sapphire-base structures versus the plate tilt. As clearly highlighted by this graph, the parallelism must be perfect to prevent power flow yielding *Q* factor limitations. For example, a HBAR built on a 4 inch wafer with a total thickness variation (TTV) of 3µm (commercially accessible for Silicon) does not suffer from any parallelism defect and therefore the quality of its resonances is almost not limited by this effect (Q>105). However, one can see that a thickness

acoustic resonance quality [19], [11]. Figure 15 shows an example of these works [19].

properties) and available as 4 inch wafers, excepted for YAG substrates.

variation of 3µm on 1cm yields effective limitation of the quality factor (*Q*<104).

and any loading due to the electrodes.

various materials [19].

The final step of the HBAR fabrication is the deposition and patterning of the top-side electrode. Aluminum electrodes are then deposited on the thinned LiNbO3 plate surface with a lift-off process. This top electrode allows for connecting the HBAR-based sensor and for characterization operations.

**Figure 14.** Flip chip of HBAR resonator on PCB substrate.

For all HBAR device, one technological problematic concerns packaging. Due to the HBAR operation, both sides must be kept free of any stress or absorbing condition. HBAR packaging therefore requires specific developments to meet such conditions. Experimental developments reveal that flip-chip techniques are the most appropriate approach in that purpose (as shown in figure 14).

## **4. HBAR optimization**

## **4.1. Minimizing losses in HBARs**

Since the 80's, HBAR devices have demonstrated high quality factor at high frequencies compared to other acoustic devices such as BAW, SAW. *Q.f* products around 1.1×1014 have already been obtained for high overtones using aluminum nitride (AlN) thin lms deposited onto sapphire [3]. Hongyu Yu *and al.* showed HBAR with a structure of 0.10µm Al /0.88µm ZnO /0.10µm Al /400µm Sapphire which was measured to have a loaded *Q* of respectively 15,000 and 19,000 for series and parallel resonant frequencies around 3.7GHz. The temperature coefficient of the resonant frequency is -28.5ppm/°C [17]. Resonators obtained by LiNbO3 wafer as a transducer bonded on another LiNbO3 wafer used as the HBAR substrate exhibit *Q* factors of 53,000 at 1.5GHz using the Gold bonding technique [5] and *Q.f* product above 8.1013 with an 800nm thickness for the piezoelectric layer by Smart Cut approach [18]. Understanding losses phenomena helps to design high quality factor devices. Loss origins can be classified into two categories: material (intrinsic) and geometry (technology-related). Due to the architecture of HBAR, the quality factor of such devices depends on the crystalline losses and on the material isotropy, on the surfaces parallelism and any loading due to the electrodes.

308 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

ready for surface processing.

for characterization operations.

purpose (as shown in figure 14).

**4.1. Minimizing losses in HBARs** 

**4. HBAR optimization** 

**Figure 14.** Flip chip of HBAR resonator on PCB substrate.

The piezoelectric wafer is subsequently thinned by lapping step to an overall thickness of 20µm. The lapping machine used in that purpose and shown in figure 13 is a SOMOS double side lapping/polishing machine based on a planetary motion of the wafers (up to 4" diameter) to promote abrasion homogeneity. An abrasive solution of silicon carbide is used here. The speed of the lapping is controlled by choosing the speed of rotation of the lapping machine stages, the load on the wafer, the rate of flow or the concentration of the abrasive. Once close to the expected thickness, the lapping process is followed by a micro-polishing step. This step uses similar equipments dedicated to polishing operation and hence using abrasive solution (colloidal silica) with smaller grain. This polishing step is applied until the average surface roughness *ra* remains larger than 3nm. Afterward, the wafer is considered

The final step of the HBAR fabrication is the deposition and patterning of the top-side electrode. Aluminum electrodes are then deposited on the thinned LiNbO3 plate surface with a lift-off process. This top electrode allows for connecting the HBAR-based sensor and

For all HBAR device, one technological problematic concerns packaging. Due to the HBAR operation, both sides must be kept free of any stress or absorbing condition. HBAR packaging therefore requires specific developments to meet such conditions. Experimental developments reveal that flip-chip techniques are the most appropriate approach in that

2mm HBAR

Since the 80's, HBAR devices have demonstrated high quality factor at high frequencies compared to other acoustic devices such as BAW, SAW. *Q.f* products around 1.1×1014 have already been obtained for high overtones using aluminum nitride (AlN) thin lms deposited onto sapphire [3]. Hongyu Yu *and al.* showed HBAR with a structure of 0.10µm Al /0.88µm ZnO /0.10µm Al /400µm Sapphire which was measured to have a loaded *Q* of respectively 15,000 and 19,000 for series and parallel resonant frequencies around 3.7GHz. The temperature coefficient of the resonant frequency is -28.5ppm/°C [17]. Resonators obtained by LiNbO3 wafer as a transducer bonded on another LiNbO3 wafer used as the HBAR

**Figure 15.** Losses per wave length and the resonator's quality *Q* as a function of frequency (GHz) for various materials [19].

As explained before, the quality factor is directly link to the acoustic quality of the substrate. Some works have already been done to compare and improve materials to favor high acoustic resonance quality [19], [11]. Figure 15 shows an example of these works [19].

The polishing process providing damaged-free ultra-smooth surfaces is essential, as well as checking the substrate quality by X-ray topography. To take into account current industrial needs, using the technology of material crystal growth is crucial to obtain large wafers. In this context, LiNbO3, LiTaO3 Sapphire, and YAG are the preferred candidates as they do present effective acoustic quality (*i.e.* reduced visco-elastic and dielectric damping properties) and available as 4 inch wafers, excepted for YAG substrates.

The defect of parallelism between two surfaces of HBAR devices dramatically limits the quality factor [11]. Figure 16 shows the quality factor of HBAR modes on Sapphire-base structures versus the plate tilt. As clearly highlighted by this graph, the parallelism must be perfect to prevent power flow yielding *Q* factor limitations. For example, a HBAR built on a 4 inch wafer with a total thickness variation (TTV) of 3µm (commercially accessible for Silicon) does not suffer from any parallelism defect and therefore the quality of its resonances is almost not limited by this effect (Q>105). However, one can see that a thickness variation of 3µm on 1cm yields effective limitation of the quality factor (*Q*<104).

High-Overtone Bulk Acoustic Resonator 311

The conditions of metal sputtering can influence the nature of the metallic electrode. Indeed, the conditions of metal sputtering for thin layers modify the density and the rate of impurity of the layer. The optimum must be found to have the highest metal density with the lowest impurities. Furthermore, some works compare the influence of different metallic layers (Al, Au, W, Ag) on the quality factor. If we consider the modified Butterworth-Van Dyke (MBVD) model, the best electrode is constituted with the lowest resistivity (Au), but experimentations also show the influence of other parameters. Thus, a Molybdenum layer

Generally speaking, low losses applications also require a temperature compensation for the resonator. One solution is to have intrinsic compensation of temperature and it is the purpose of the next paragraph. Another solution is to control frequency by measuring

One challenge of the radio-frequency bulk acoustic devices is the temperature stability of their resonance frequency. A lot of work has been achieved exploiting thin piezoelectric films for developing temperature-compensated HBARs, with various successes. The possibility to use single crystal thinned films appears as an alternative to control the piezoelectric film properties (velocity, coupling, temperature sensitivity, and so on.) and to globally reconsider material association according to the technological assembly process

The celebrated Campbell&Jones method [25] is used here for predicting the Temperature Coefficient of Frequency (TCF) of any mode of a given HBAR. As it has been reported

> <sup>2</sup> *<sup>v</sup> df dv de <sup>f</sup> T TT ef v e*

Which means that the frequency changes due to temperature variations is computed as the difference between the development of the velocity and of the stack thickness versus temperature. Theoretically using a standard anisotropic 1D model reveals that zero temperature coefficients of frequency (TCF) can be obtained and optimized along the mode order. It is well-known that Quartz and fused Silica (glass) do exhibit positive TCFs. So the use of the other temperature-compensated Quartz orientations, and hence of any other material sharing such property, has been checked theoretically and reveals applicable as

As example, Lithium of Niobate and Quartz have been associated for the fabrication of shear-wave based HBARs. LiNbO3 provides crystal orientations for which very strongly coupled shear waves exist (*ks²* in excess of 45%) whereas *AT* cut of Quartz allows for

and *T* are respectively frequency, thickness of resonator, wave velocity and

(1)

hundred times in previous papers, only the main basic equation is reported below:

used as an electrode shows better results due to better acoustic impedance [23].

temperature.

previously presented.

*f*, *e*, 

well.

temperature.

**4.2. Temperature compensation** 

**Figure 16.** Parallelism-limited *Q* in a single-port resonator built on Z cut Sapphire substrates [11].

The shape, size and nature of the electrodes can be also important to manufacture high *Q* HBAR devices. Some works have been done on electrodes of HBAR devices [20], [21], [22], [23].

**Figure 17.** Experimental and Modeled Unloaded Q versus aperture [20].

D. S. Bailey *and al.* showed that the HBAR does not follow the one dimensional computer model [20]. Indeed, figure 17 shows the difference between the experimental *Q* and the theoretical *Q* versus the aperture of the electrode. The difference is due to the diffraction effect. The optimum electrode area can depend on two main parameters: the clamp capacitance *C0* and the geometry. This capacitance *C0* is proportional to the surface area and influences other parameters of resonator such as difference in impedance at series and resonance frequency. With a large active area, defects in transducer crystal or of geometry can happen more easily. The optimization of the area shape and surface to limit the diffraction effect and improve the quality factor is an area of ongoing works.

Furthermore, for ultra-high frequency HBAR devices, the electrodes are not thin compared to transducer layer. The thickness and the nature of electrodes have an influence on the quality factor and the other parameters such as the electromechanical coupling coefficient and the resonance frequency. Many works have been done on this subject [23], [24], [21].

The conditions of metal sputtering can influence the nature of the metallic electrode. Indeed, the conditions of metal sputtering for thin layers modify the density and the rate of impurity of the layer. The optimum must be found to have the highest metal density with the lowest impurities. Furthermore, some works compare the influence of different metallic layers (Al, Au, W, Ag) on the quality factor. If we consider the modified Butterworth-Van Dyke (MBVD) model, the best electrode is constituted with the lowest resistivity (Au), but experimentations also show the influence of other parameters. Thus, a Molybdenum layer used as an electrode shows better results due to better acoustic impedance [23].

Generally speaking, low losses applications also require a temperature compensation for the resonator. One solution is to have intrinsic compensation of temperature and it is the purpose of the next paragraph. Another solution is to control frequency by measuring temperature.

## **4.2. Temperature compensation**

310 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

**Figure 16.** Parallelism-limited *Q* in a single-port resonator built on Z cut Sapphire substrates [11].

devices. Some works have been done on electrodes of HBAR devices [20], [21], [22], [23].

**Figure 17.** Experimental and Modeled Unloaded Q versus aperture [20].

diffraction effect and improve the quality factor is an area of ongoing works.

The shape, size and nature of the electrodes can be also important to manufacture high *Q* HBAR

D. S. Bailey *and al.* showed that the HBAR does not follow the one dimensional computer model [20]. Indeed, figure 17 shows the difference between the experimental *Q* and the theoretical *Q* versus the aperture of the electrode. The difference is due to the diffraction effect. The optimum electrode area can depend on two main parameters: the clamp capacitance *C0* and the geometry. This capacitance *C0* is proportional to the surface area and influences other parameters of resonator such as difference in impedance at series and resonance frequency. With a large active area, defects in transducer crystal or of geometry can happen more easily. The optimization of the area shape and surface to limit the

Furthermore, for ultra-high frequency HBAR devices, the electrodes are not thin compared to transducer layer. The thickness and the nature of electrodes have an influence on the quality factor and the other parameters such as the electromechanical coupling coefficient and the resonance frequency. Many works have been done on this subject [23], [24], [21].

One challenge of the radio-frequency bulk acoustic devices is the temperature stability of their resonance frequency. A lot of work has been achieved exploiting thin piezoelectric films for developing temperature-compensated HBARs, with various successes. The possibility to use single crystal thinned films appears as an alternative to control the piezoelectric film properties (velocity, coupling, temperature sensitivity, and so on.) and to globally reconsider material association according to the technological assembly process previously presented.

The celebrated Campbell&Jones method [25] is used here for predicting the Temperature Coefficient of Frequency (TCF) of any mode of a given HBAR. As it has been reported hundred times in previous papers, only the main basic equation is reported below:

$$f = \frac{v}{2e} \to \frac{df}{f}(T) = \frac{dv}{v}(T) - \frac{de}{e}(T) \tag{1}$$

*f*, *e*, and *T* are respectively frequency, thickness of resonator, wave velocity and temperature.

Which means that the frequency changes due to temperature variations is computed as the difference between the development of the velocity and of the stack thickness versus temperature. Theoretically using a standard anisotropic 1D model reveals that zero temperature coefficients of frequency (TCF) can be obtained and optimized along the mode order. It is well-known that Quartz and fused Silica (glass) do exhibit positive TCFs. So the use of the other temperature-compensated Quartz orientations, and hence of any other material sharing such property, has been checked theoretically and reveals applicable as well.

As example, Lithium of Niobate and Quartz have been associated for the fabrication of shear-wave based HBARs. LiNbO3 provides crystal orientations for which very strongly coupled shear waves exist (*ks²* in excess of 45%) whereas *AT* cut of Quartz allows for

compensating second order frequency-temperature effects [WO2009156658 (A1)]. Although this idea was already proposed using other material combinations [US Patent #3401275A], no real design process was presented until now and therefore the possibility to actually determine structures allowing for high frequency operation with first order TCF smaller than 1ppm.K-1 was quite hypothetical, but improvement of numerical tools allows this design.

High-Overtone Bulk Acoustic Resonator 313

numerical analysis and that if an intuitive approach allows for a first order definition of crystal orientations, the complicated distribution of energy within the stack versus all the

**Figure 19.** Plot of TCF of a HBAR built on a (YX*l*)/163° LiNbO3 thinned plate bonded on (YX*lt*)/36°/90° Quartz substrate for various Lithium of Niobate/Quartz thickness ratio (Quartz thickness arbitrary fixed

Industrial acoustic-resonator-based sensors require adapted electronics to be efficiently


with low harmonic generation. BAW, SAW and FBAR can use this electronic. - The second way is to have electronic interrogation which finds frequency resonance in determined range of frequency. With classic method it is possible to obtain 100Hz of resolution for 434MHz sensors [28]. If electronics is improving, we can achieve 5Hz of resolution for 434MHz sensors [29]. In this case, clock of electronic is really important for performance. All kind of resonators sensors can be interrogated by this technique,

especially HBAR device which present high overtone generation.

operated. Two main approaches have been developed in that purpose:

structure parameters induces more intrication in the design process.

to 50 µm) [27].

**5. HBAR applications** 

**5.1. HBAR sensors** 

Nevertheless, some works show the possibility to have an intrinsic compensation of the temperature for HBAR devices [8], [26]. Figure 18 shows the temperature dependence for different configuration of HBAR devices constituted by LiNbO3 and Quartz layers with different cut orientations. This work shows clearly that the choice of materials and the cut orientation of these materials have a direct impact on the frequency shift with temperature variations [26].

**Figure 18.** Electrical responses for two different configurations of HBAR (left), frequency variation versus temperature for six configurations of HBAR (right) [26].

Moreover, the frequency dependence on temperature is different for each overtone of HBAR devices. The thickness ratio between the transducer layer and the substrate also influences the frequency variations with different temperatures. Figure 19 shows the computation of the temperature coefficient of frequency (TCF) of a HBAR for various Lithium of Niobate / Quartz thickness ratios. This HBAR device is built on a (YX*l*)/163° LiNbO3 thinned plate bonded on (YX*lt*)/36°/90° Quartz substrate of 50µm [27]. One can see that depending on the harmonic number, the *TCF1* changes from +1 to -14ppm.K-1. Furthermore, depending on the harmonic of the transducer alone, the *TCF1* may notably change and thus it cannot be considered as a simple periodic function versus harmonic number. Therefore, it is mandatory to accurately consider all the actual features of the structure for an accurate design of a resonator, *i.e.* the operation frequency, the harmonic number and the thickness ratio for a given structure. To complete this, one should also account for the actual thickness of the device as this parameter will control the possibility to select one (frequency/harmonic number) couple. Finally, it clearly appears that the analysis of such HBAR TCF requires a numerical analysis and that if an intuitive approach allows for a first order definition of crystal orientations, the complicated distribution of energy within the stack versus all the structure parameters induces more intrication in the design process.

**Figure 19.** Plot of TCF of a HBAR built on a (YX*l*)/163° LiNbO3 thinned plate bonded on (YX*lt*)/36°/90° Quartz substrate for various Lithium of Niobate/Quartz thickness ratio (Quartz thickness arbitrary fixed to 50 µm) [27].

## **5. HBAR applications**

### **5.1. HBAR sensors**

312 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

design.

variations [26].

compensating second order frequency-temperature effects [WO2009156658 (A1)]. Although this idea was already proposed using other material combinations [US Patent #3401275A], no real design process was presented until now and therefore the possibility to actually determine structures allowing for high frequency operation with first order TCF smaller than 1ppm.K-1 was quite hypothetical, but improvement of numerical tools allows this

Nevertheless, some works show the possibility to have an intrinsic compensation of the temperature for HBAR devices [8], [26]. Figure 18 shows the temperature dependence for different configuration of HBAR devices constituted by LiNbO3 and Quartz layers with different cut orientations. This work shows clearly that the choice of materials and the cut orientation of these materials have a direct impact on the frequency shift with temperature

**Figure 18.** Electrical responses for two different configurations of HBAR (left), frequency variation

Moreover, the frequency dependence on temperature is different for each overtone of HBAR devices. The thickness ratio between the transducer layer and the substrate also influences the frequency variations with different temperatures. Figure 19 shows the computation of the temperature coefficient of frequency (TCF) of a HBAR for various Lithium of Niobate / Quartz thickness ratios. This HBAR device is built on a (YX*l*)/163° LiNbO3 thinned plate bonded on (YX*lt*)/36°/90° Quartz substrate of 50µm [27]. One can see that depending on the harmonic number, the *TCF1* changes from +1 to -14ppm.K-1. Furthermore, depending on the harmonic of the transducer alone, the *TCF1* may notably change and thus it cannot be considered as a simple periodic function versus harmonic number. Therefore, it is mandatory to accurately consider all the actual features of the structure for an accurate design of a resonator, *i.e.* the operation frequency, the harmonic number and the thickness ratio for a given structure. To complete this, one should also account for the actual thickness of the device as this parameter will control the possibility to select one (frequency/harmonic number) couple. Finally, it clearly appears that the analysis of such HBAR TCF requires a

versus temperature for six configurations of HBAR (right) [26].

Industrial acoustic-resonator-based sensors require adapted electronics to be efficiently operated. Two main approaches have been developed in that purpose:


Acoustic sensor is passive sensor. Device combine with antenna could be having great interest. Indeed, electromagnetic waves can be changed on electrical waves on electrodes, which can excite acoustic waves by piezoelectric effect. Furthermore this phenomenon is linear and invertible. So, wireless interrogation is possible with acoustic sensors. Wireless communication presents great interest for all hard environments. In that way, acoustic sensors can be use in engine, close environment and more generally in all environments where wire can not be employed.

High-Overtone Bulk Acoustic Resonator 315

micro-cavity then plays the role of a mirror for the waves. The structure of such device is shown in figure 20. The surface of the cavity should at minimum coincide strictly to the surface of the transducer, but to ease the fabrication (particularly to manage alignment issues) the cavity largely overlaps the transducer aperture. The micro-cavity/micro-mirror could be placed at different deph into the HBAR stack. Its location will define the HBAR

**Figure 20.** HBAR structures presented low frequency sensitive to stress (a), and highly frequency

A lot of works has been done on liquid or gaz HBAR / FBAR sensor and as most representative example, gravimetric sensor. The basic principle of gravimetric acoustic wave sensors is the measurement of the phase velocity variations due to an adsorbed mass or a layer thickness change atop the device during a chemical reaction: this phase velocity is dependent on the boundary conditions of the propagating acoustic wave and is affected either by the layer properties or its thickness. The usual principle exploits bulk acoustic waves, yielding the well-known concept of Quartz Crystal Micro-balance (QCM). The gravimetric sensitivity of the QCM is directly related to its thickness and as a consequence to its fundamental frequency *f0*. Adsorption on one side of the resonator modifies its resonance conditions and thus allows for a gravimetric detection. Furthermore, it is possible to functionalize the surface with specific reactants to provide information on the concentration of the adsorbed (target) species in the medium surrounding the sensor [31]. The case of HBAR is particularly attractive as one can expect probing the adsorbed material at various frequencies,

Copper electro-deposition on the back side of a HBAR has been used for calibrating the gravimetric sensitivity of its overtones. This approach was particularly implemented as it provides an independent estimate of the deposited metal mass through the measurement of the current. A negative current indicates copper reduction (deposition on the working electrode) whereas a positive current indicates oxidation (copper removal from the working electrode). Simultaneous to the current monitoring, the acoustic phase and magnitude at fixed frequency are recorded [30]. The figure 21.a shows four different overtone frequencies (red dot) recorded. Figure 21.b. shows relative frequency variations and clearly shows the sensitivity difference of the four HBAR overtones. Sensitivity of gravimetric HBAR directly

sensitive to stress with the realization of micro-mirror under the transducer aperture [8].

providing frequency-dependent information such as viscosity for instance.

sensibility to stress [8].

With wireless interrogation, antenna size, quality factor of resonator, frequency have a strong impact. With increasing of frequency, antenna size decrease. Indeed, the size of antenna is equal to the quarter of wavelength. When higher ISM band is used, quality factor need to be increase to give the same obstruction of the ISM bandwidth. At -3dB, the bandwidth of resonator is proportional to frequency divided by quality factor. And finally, the flight time is proportional to quality factor divided by frequency. They are two consequences of this flight time. Firstly, to have enough energy when frequency increase, the quality factor need to increase. As example, SAW resonators at 434MHz ISM band have quality factor of 10,000 and can be interrogated by wireless approach. To pass at 2.45GHz ISM band, a quality factor equal to 20,000 is required. HBAR devices achieve these characteristics. Secondly, refresh rate increases with frequency. With bandwidth of few kHz the refresh rate is around one millisecond. In this case, quality factor of sensor could not be too higher. So, quality factor of HBAR device need to be optimize for wireless sensor application.

HBAR devices present a great advantage for achieving sensors device. As previously discussed, frequency shift due to temperature effects can be minimized and even compensated, but also magnified as well. As a consequence, HBAR temperature sensors are considered first. Moreover, due to high number of overtones of such devices, it is also possible to develop sensors exhibiting different sensitivity to a given parametric effect at different frequencies. Acoustic devices can also be effectively exploited as stress sensor or pressure sensors. The fabrication of SAW pressure sensor based on thinned Quartz membrane (for instance) was strongly investigated due to the dependence of the wave velocity versus tensile stress at the surface of the membrane when bent by pressure. In the case of bulk wave propagating in such a membrane, the strain variations across the membrane thickness forbid the use of such an approach to develop pressure sensor applications. This can be easily demonstrated using for instance static finite element analysis with a very simple mesh. Indeed, the strain and hence the stress change their signs along the membrane thickness when submitted to pressure. As a consequence, the strain variation across the HBAR generates equilibration of the velocity variations. On the one hand, the strain below the membrane neutral line yields an increase of resonant frequency of the HBAR; on the other hand, the strain above the neutral line yields a decrease of this frequency. Consequently, the resulting frequency shift is negligible. One solution consists in the fabrication of a micro-cavity within the HBAR stack near the neutral line. If the transducer of the HBAR structure is straight above this micro-cavity, the emitted bulk waves are reflected by this micro-cavity and hence confined in this membrane location. The micro-cavity then plays the role of a mirror for the waves. The structure of such device is shown in figure 20. The surface of the cavity should at minimum coincide strictly to the surface of the transducer, but to ease the fabrication (particularly to manage alignment issues) the cavity largely overlaps the transducer aperture. The micro-cavity/micro-mirror could be placed at different deph into the HBAR stack. Its location will define the HBAR sensibility to stress [8].

314 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

where wire can not be employed.

application.

Acoustic sensor is passive sensor. Device combine with antenna could be having great interest. Indeed, electromagnetic waves can be changed on electrical waves on electrodes, which can excite acoustic waves by piezoelectric effect. Furthermore this phenomenon is linear and invertible. So, wireless interrogation is possible with acoustic sensors. Wireless communication presents great interest for all hard environments. In that way, acoustic sensors can be use in engine, close environment and more generally in all environments

With wireless interrogation, antenna size, quality factor of resonator, frequency have a strong impact. With increasing of frequency, antenna size decrease. Indeed, the size of antenna is equal to the quarter of wavelength. When higher ISM band is used, quality factor need to be increase to give the same obstruction of the ISM bandwidth. At -3dB, the bandwidth of resonator is proportional to frequency divided by quality factor. And finally, the flight time is proportional to quality factor divided by frequency. They are two consequences of this flight time. Firstly, to have enough energy when frequency increase, the quality factor need to increase. As example, SAW resonators at 434MHz ISM band have quality factor of 10,000 and can be interrogated by wireless approach. To pass at 2.45GHz ISM band, a quality factor equal to 20,000 is required. HBAR devices achieve these characteristics. Secondly, refresh rate increases with frequency. With bandwidth of few kHz the refresh rate is around one millisecond. In this case, quality factor of sensor could not be too higher. So, quality factor of HBAR device need to be optimize for wireless sensor

HBAR devices present a great advantage for achieving sensors device. As previously discussed, frequency shift due to temperature effects can be minimized and even compensated, but also magnified as well. As a consequence, HBAR temperature sensors are considered first. Moreover, due to high number of overtones of such devices, it is also possible to develop sensors exhibiting different sensitivity to a given parametric effect at different frequencies. Acoustic devices can also be effectively exploited as stress sensor or pressure sensors. The fabrication of SAW pressure sensor based on thinned Quartz membrane (for instance) was strongly investigated due to the dependence of the wave velocity versus tensile stress at the surface of the membrane when bent by pressure. In the case of bulk wave propagating in such a membrane, the strain variations across the membrane thickness forbid the use of such an approach to develop pressure sensor applications. This can be easily demonstrated using for instance static finite element analysis with a very simple mesh. Indeed, the strain and hence the stress change their signs along the membrane thickness when submitted to pressure. As a consequence, the strain variation across the HBAR generates equilibration of the velocity variations. On the one hand, the strain below the membrane neutral line yields an increase of resonant frequency of the HBAR; on the other hand, the strain above the neutral line yields a decrease of this frequency. Consequently, the resulting frequency shift is negligible. One solution consists in the fabrication of a micro-cavity within the HBAR stack near the neutral line. If the transducer of the HBAR structure is straight above this micro-cavity, the emitted bulk waves are reflected by this micro-cavity and hence confined in this membrane location. The

**Figure 20.** HBAR structures presented low frequency sensitive to stress (a), and highly frequency sensitive to stress with the realization of micro-mirror under the transducer aperture [8].

A lot of works has been done on liquid or gaz HBAR / FBAR sensor and as most representative example, gravimetric sensor. The basic principle of gravimetric acoustic wave sensors is the measurement of the phase velocity variations due to an adsorbed mass or a layer thickness change atop the device during a chemical reaction: this phase velocity is dependent on the boundary conditions of the propagating acoustic wave and is affected either by the layer properties or its thickness. The usual principle exploits bulk acoustic waves, yielding the well-known concept of Quartz Crystal Micro-balance (QCM). The gravimetric sensitivity of the QCM is directly related to its thickness and as a consequence to its fundamental frequency *f0*. Adsorption on one side of the resonator modifies its resonance conditions and thus allows for a gravimetric detection. Furthermore, it is possible to functionalize the surface with specific reactants to provide information on the concentration of the adsorbed (target) species in the medium surrounding the sensor [31]. The case of HBAR is particularly attractive as one can expect probing the adsorbed material at various frequencies, providing frequency-dependent information such as viscosity for instance.

Copper electro-deposition on the back side of a HBAR has been used for calibrating the gravimetric sensitivity of its overtones. This approach was particularly implemented as it provides an independent estimate of the deposited metal mass through the measurement of the current. A negative current indicates copper reduction (deposition on the working electrode) whereas a positive current indicates oxidation (copper removal from the working electrode). Simultaneous to the current monitoring, the acoustic phase and magnitude at fixed frequency are recorded [30]. The figure 21.a shows four different overtone frequencies (red dot) recorded. Figure 21.b. shows relative frequency variations and clearly shows the sensitivity difference of the four HBAR overtones. Sensitivity of gravimetric HBAR directly

depends on the stack thickness and more precisely on copper thickness versus transducer thickness. The best gravimetric HBAR sensor is constituted by the thinnest stack with metallic thickness equal to quarter of wavelength [31].

High-Overtone Bulk Acoustic Resonator 317

HBAR then are capable to address high frequency source applications without requiring multiplication stages as usually achieved. The idea then is to evaluate the effective interest of HBAR for direct frequency synthesis, reducing the oscillator architecture complexity and

Hongyu Yu and *al.* have presented a local oscillator based on a HBAR resonator associated to an atomic clock [34]. Atomic clocks are used for embedded applications which need high stability performance such as GPS station. The atomic transition allows having long-term stability in this application, but it presents poor short-term stability. To success oscillator based on this atomic transition, local oscillator is needed. This local oscillator stabilizes the short-term variation of the global oscillator with its short-term performance. The local oscillator need to have good phase noise (better than -70dBc/Hz at 1kHz for instance) to prevent global degradation of clock stability. Figure 22 shows the phase noise measurement data of the 3.67GHz Pierce oscillator and the 1.2GHz Colpitts oscillator, and the Allan deviation of the free-running 3.67GHz oscillator that consumes only about 3mW [34]. Local oscillator based on HBAR resonator need frequency control to achieve the atomic transition frequency. With the modulation of the HBAR frequency with an external synthesizer and FBAR filters, the local oscillator locked to the coherent population trapping resonance.

**Figure 22.** Phase noise measurement data of (a) the 3.67GHz Pierce oscillator and (b) the 1.2GHz Colpitts oscillator. (c) Allan deviation of the free-running 3.67GHz oscillator that consumes only about

Other applications require low phase noise oscillator such as embedded RADAR. A radiofrequency oscillator operating near the 434MHz-centered ISM band validates the capability of the above-mentioned HBAR for such purposes. The composite substrates have been built using 3-inches (YX*l*)/163° LiNbO3 cut wafer bonded and thinned down to 15µm onto a 350µm thick (YX*lt*)/34°/90° Quartz base. Single-port resonators operating near 434MHz (exploiting the third harmonic of the thinned Lithium of Niobate plate as the HBAR "motor") have been then manufactured. Electrical and thermoelectric characterizations have shown quality factor of the resonance in excess of 20,000, yielding a *Q.f* product of about 1013 and a third order frequency-temperature behavior. A SAW filter was used to select the ISM band and to filter the high spectral density HBAR response (figure 23). The oscillator then has been measured using a phase noise automatic bench. A phase noise better than -- 160dBc/Hz at 100kHz has been measured as well as a -165dBc/Hz level at 1MHz from the

3mW [34].

potentially improving the corresponding operational features.

**Figure 21.** (a) Overtones of the fundamental transducer layer mode of a gravimetric HBAR sensor with four probed frequencies (red dot), (b) relative frequency variations of these four frequencies with their relative sensitivities [30].

## **5.2. HBAR-stabilized oscillators**

Radio-Frequency oscillators can be stabilized by various resonating devices. Their stability is mainly conditioned by the spectral quality of the resonator even if the oscillator loop electronics must be optimized to lower the generated noise as much as possible. For midterm stability, temperature compensation is a key point and allows for notably improving the corresponding figures of merit. The possibility to build temperature compensated HBARs has been shown in paragraph 4.2 and is a key-point for the fabrication of oscillator exhibiting short-term stability compatible with practical applications.

Moreover, the frequency stability of an oscillator can be characterized by its single-sideband phase noise, *L{fm}*. Leeson's equation [33] shows that low phase noise operation can be achieved by increasing the loaded quality factor *Qload* of the resonator. According to Leeson's model, a high resonator quality factor (*Q*) or circulating power level improves the phase noise and, therefore, the short-term stability of the oscillator. Considering these aspects, HBAR device features for the frequency stability reveal more favourable than FBAR or SAW device ones. Therefore HBAR should allow for notably improving oscillator performances. However, multi-overtone features of HBAR do not facilitate resonance lock for oscillator applications. Therefore SAW or FBAR device are generally used to filter the frequency of HBAR. Consequently, the compactness of HBAR is deteriorated due to the need for this filter. Optimizing HBAR spectral response then is still an open question and should receive more attention in future developments. One can note that using single port HBARs with optimized frequency separation between the overtones [32] may allow to get rid of this filtering operation.

HBAR then are capable to address high frequency source applications without requiring multiplication stages as usually achieved. The idea then is to evaluate the effective interest of HBAR for direct frequency synthesis, reducing the oscillator architecture complexity and potentially improving the corresponding operational features.

316 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

metallic thickness equal to quarter of wavelength [31].

relative sensitivities [30].

filtering operation.

**5.2. HBAR-stabilized oscillators** 

depends on the stack thickness and more precisely on copper thickness versus transducer thickness. The best gravimetric HBAR sensor is constituted by the thinnest stack with

**Figure 21.** (a) Overtones of the fundamental transducer layer mode of a gravimetric HBAR sensor with four probed frequencies (red dot), (b) relative frequency variations of these four frequencies with their

Radio-Frequency oscillators can be stabilized by various resonating devices. Their stability is mainly conditioned by the spectral quality of the resonator even if the oscillator loop electronics must be optimized to lower the generated noise as much as possible. For midterm stability, temperature compensation is a key point and allows for notably improving the corresponding figures of merit. The possibility to build temperature compensated HBARs has been shown in paragraph 4.2 and is a key-point for the fabrication of oscillator

Moreover, the frequency stability of an oscillator can be characterized by its single-sideband phase noise, *L{fm}*. Leeson's equation [33] shows that low phase noise operation can be achieved by increasing the loaded quality factor *Qload* of the resonator. According to Leeson's model, a high resonator quality factor (*Q*) or circulating power level improves the phase noise and, therefore, the short-term stability of the oscillator. Considering these aspects, HBAR device features for the frequency stability reveal more favourable than FBAR or SAW device ones. Therefore HBAR should allow for notably improving oscillator performances. However, multi-overtone features of HBAR do not facilitate resonance lock for oscillator applications. Therefore SAW or FBAR device are generally used to filter the frequency of HBAR. Consequently, the compactness of HBAR is deteriorated due to the need for this filter. Optimizing HBAR spectral response then is still an open question and should receive more attention in future developments. One can note that using single port HBARs with optimized frequency separation between the overtones [32] may allow to get rid of this

exhibiting short-term stability compatible with practical applications.

Hongyu Yu and *al.* have presented a local oscillator based on a HBAR resonator associated to an atomic clock [34]. Atomic clocks are used for embedded applications which need high stability performance such as GPS station. The atomic transition allows having long-term stability in this application, but it presents poor short-term stability. To success oscillator based on this atomic transition, local oscillator is needed. This local oscillator stabilizes the short-term variation of the global oscillator with its short-term performance. The local oscillator need to have good phase noise (better than -70dBc/Hz at 1kHz for instance) to prevent global degradation of clock stability. Figure 22 shows the phase noise measurement data of the 3.67GHz Pierce oscillator and the 1.2GHz Colpitts oscillator, and the Allan deviation of the free-running 3.67GHz oscillator that consumes only about 3mW [34]. Local oscillator based on HBAR resonator need frequency control to achieve the atomic transition frequency. With the modulation of the HBAR frequency with an external synthesizer and FBAR filters, the local oscillator locked to the coherent population trapping resonance.

**Figure 22.** Phase noise measurement data of (a) the 3.67GHz Pierce oscillator and (b) the 1.2GHz Colpitts oscillator. (c) Allan deviation of the free-running 3.67GHz oscillator that consumes only about 3mW [34].

Other applications require low phase noise oscillator such as embedded RADAR. A radiofrequency oscillator operating near the 434MHz-centered ISM band validates the capability of the above-mentioned HBAR for such purposes. The composite substrates have been built using 3-inches (YX*l*)/163° LiNbO3 cut wafer bonded and thinned down to 15µm onto a 350µm thick (YX*lt*)/34°/90° Quartz base. Single-port resonators operating near 434MHz (exploiting the third harmonic of the thinned Lithium of Niobate plate as the HBAR "motor") have been then manufactured. Electrical and thermoelectric characterizations have shown quality factor of the resonance in excess of 20,000, yielding a *Q.f* product of about 1013 and a third order frequency-temperature behavior. A SAW filter was used to select the ISM band and to filter the high spectral density HBAR response (figure 23). The oscillator then has been measured using a phase noise automatic bench. A phase noise better than -- 160dBc/Hz at 100kHz has been measured as well as a -165dBc/Hz level at 1MHz from the

career (figure 23). Short-term stability characterizations show that the resonator stability is better than 10-9 at room conditions (no temperature stabilization).

High-Overtone Bulk Acoustic Resonator 319

**Figure 24.** Phase noise curves for oscillators at 935MHz, 1,636MHz compared with phase noise of an

HBAR have been developed for the fabrication of passive radio-frequency devices capable to overcome standard SAW and BAW limitations considering the quality of the resonance,

These devices actually maximize the *Q* factor that can be obtained at room temperature using elastic waves, yielding quality factor times frequency products (*Q.f*) close or slightly above 1014, *i.e.* effective *Q* factors of about 10,000 at 10GHz in theory (practically, *Q* factors in excess of 50,000 between 1.5 and 2GHz were experimentally achieved). Single-port or fourport resonator has been described and the different approach of manufacturing as been explained. The choice of acoustic wave and acoustic substrate permits to address large range of application. First one is chose value of the quality factor and the electromechanical coupling coefficient of overtone frequencies. Second one is to minimize frequency shift due to temperature variation for chosen frequency. All these possibilities allow us to address

Although HBAR device knows since several decades, HBAR device has not yet achieve is development maturity. Futures works will concern improvement of fabrication, frequency control, and wireless sensors. The large number of the parameters for optimizing HBAR in function of the applications requires well-control generic process of fabrication. The selection of the frequency resonance is also a key point for the emergence of HBAR devices, as the frequency tuning. And finally, wireless HBAR sensor need strong effort of

Some simple applications and large potentiality of HBAR conception has been presented. Two main approaches exist for the realization of HBAR devices. The first one based on

oscillator stabilized by a classical BAW resonator at 100MHz [33].

complexity of technological fabrication and operation frequencies.

different applications such as sensor or low phase noise applications.

**6. Conclusions and perspectives** 

development.

**Figure 23.** Phase noise curves for oscillators at 434MHz constituted by High Overtone Bulk Resonator, SAW filter and Colpitts electronics.

To achieve higher frequency (above 1GHz), four-port resonators are mandatory due to the difficulty to adjust the oscillator tuning elements. For this application, the temperature stability is required and therefore the resonator exploits a (YX*l*)/163° LiNbO3 thinned layer atop a (YX*lt*)/36°/90° Quartz substrate. The electrodes defining the coupled transducers (four-port resonator) are two half-circles (300µm diameter) separated by a gap of 20µm, yielding favorable conditions for using the resonator to stabilize an oscillator loop at 935MHz and 1.636GHz. The device were cut and packaged, mounted in an oscillator loop, and measurements of phase noise were performed. Figure 24 shows the phase noise of the two corresponding oscillators compared with the phase noise of an oscillator stabilized by a classical BAW resonator at 100MHz, "octar 507X100" from AR-Electronics. The oscillator near 1.6GHz clearly shows better performances than the one at 935MHz, with a noise level lower than -130dBc/Hz at 10kHz from the carrier. In order to compare the 100MHz oscillator with the 1.6GHz one, the low frequency source has to be multiplied by 16, *i.e.* +12dB must be added to the noise level. It gives a level of -140dBc/Hz at 10kHz which is not far from HBAR solution.

**Figure 24.** Phase noise curves for oscillators at 935MHz, 1,636MHz compared with phase noise of an oscillator stabilized by a classical BAW resonator at 100MHz [33].

## **6. Conclusions and perspectives**

318 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

better than 10-9 at room conditions (no temperature stabilization).

career (figure 23). Short-term stability characterizations show that the resonator stability is

**Figure 23.** Phase noise curves for oscillators at 434MHz constituted by High Overtone Bulk Resonator,

To achieve higher frequency (above 1GHz), four-port resonators are mandatory due to the difficulty to adjust the oscillator tuning elements. For this application, the temperature stability is required and therefore the resonator exploits a (YX*l*)/163° LiNbO3 thinned layer atop a (YX*lt*)/36°/90° Quartz substrate. The electrodes defining the coupled transducers (four-port resonator) are two half-circles (300µm diameter) separated by a gap of 20µm, yielding favorable conditions for using the resonator to stabilize an oscillator loop at 935MHz and 1.636GHz. The device were cut and packaged, mounted in an oscillator loop, and measurements of phase noise were performed. Figure 24 shows the phase noise of the two corresponding oscillators compared with the phase noise of an oscillator stabilized by a classical BAW resonator at 100MHz, "octar 507X100" from AR-Electronics. The oscillator near 1.6GHz clearly shows better performances than the one at 935MHz, with a noise level lower than -130dBc/Hz at 10kHz from the carrier. In order to compare the 100MHz oscillator with the 1.6GHz one, the low frequency source has to be multiplied by 16, *i.e.* +12dB must be added to the noise level. It gives a level of -140dBc/Hz at 10kHz which is not far from HBAR

SAW filter and Colpitts electronics.

solution.

HBAR have been developed for the fabrication of passive radio-frequency devices capable to overcome standard SAW and BAW limitations considering the quality of the resonance, complexity of technological fabrication and operation frequencies.

These devices actually maximize the *Q* factor that can be obtained at room temperature using elastic waves, yielding quality factor times frequency products (*Q.f*) close or slightly above 1014, *i.e.* effective *Q* factors of about 10,000 at 10GHz in theory (practically, *Q* factors in excess of 50,000 between 1.5 and 2GHz were experimentally achieved). Single-port or fourport resonator has been described and the different approach of manufacturing as been explained. The choice of acoustic wave and acoustic substrate permits to address large range of application. First one is chose value of the quality factor and the electromechanical coupling coefficient of overtone frequencies. Second one is to minimize frequency shift due to temperature variation for chosen frequency. All these possibilities allow us to address different applications such as sensor or low phase noise applications.

Although HBAR device knows since several decades, HBAR device has not yet achieve is development maturity. Futures works will concern improvement of fabrication, frequency control, and wireless sensors. The large number of the parameters for optimizing HBAR in function of the applications requires well-control generic process of fabrication. The selection of the frequency resonance is also a key point for the emergence of HBAR devices, as the frequency tuning. And finally, wireless HBAR sensor need strong effort of development.

Some simple applications and large potentiality of HBAR conception has been presented. Two main approaches exist for the realization of HBAR devices. The first one based on

piezoelectric deposition gives easily high quality factor and frequency device. The largest potentiality of conception to address different applications is obtained by the second approach based on mono-crystal wafer assembled. Further developments required to promote the industrial exploitation.

High-Overtone Bulk Acoustic Resonator 321

[6] Curran Daniel R & Al, US Patent #3401275A, 1968-09-10

*Symposium*, pp. 2040-2043, 2010.

*Forum (FCS)*, 2011

*JNTE* 2010

*IEEE International.* pp. 1-4. 2008

*(2008)* pp. 201 – 204, 2008

[7] M. Pijolat, D. Mercier, A. Reinhardt, E. Defaÿ, C. Deguet, M. Aïd, J.S. Moulet, B. Ghyselen, S. Ballandras, Mode conversion in high overtone bulk acoustic wave

[8] T. Baron, E. Lebrasseur, J.P. Romand, S. Alzuaga, S. Queste, G. Martin, D. Gachon, T. Laroche, S. Ballandras, J. Masson, Temperature compensated radio-frequency harmonic bulk acoustic resonators pressure sensors. *Proc. of the IEEE International Ultrasonics* 

[9] D. Gachon, T. Baron, G. Martin, E. Lebrasseur, E. Courjon, F. Bassignot, S. Ballandras, Laterally coupled narrow-band high overtone bulk wave filters using thinned single crystal lithium niobate layers. *Frequency Control and the European Frequency and Time* 

[11] R.A. Moore, J.T. Haynes, B.R. McAvoy. High Overtone Bulk Resonator Stabilized

[12] B. Aspar, H. Moriceau, *et al.*. The generic nature of the Smart-Cut process for thin film

[13] T. Pastureaud, M. Solal, B. Biasse, B. Aspar, J.B. Briot, W. Daniau, W. Steichen, R. Lardat, V. Laude, A. Laëns, J.M. Frietd, S. Ballandras, High Frequency Surface Acoustic Waves Excited on Thin Oriented LiNbO3 Single Crystal Layers Transferred Onto

[14] J.C. Ponçot, P. Nyeki, Ph. Defranould, J.P. Huignard. 3 GHz Bandwidth Bragg Cells.

[15] T. Baron, E. Lebrasseur, *et al.*. Development of Composite Single Crystal Wafers for Sources and Sensor applications exploiting High-overtone Bulk Acoustic Resonators.

[16] J.-S. Moulet, M. Pijolat, J. Dechamp, F. Mazen, A. Tauzin, F. Rieutord, A. Reinhardt, E. Defay, C. Deguet, B. Ghyselen, L. Clavelier, M. Aid, S. Ballandras, C. Mazure. High piezoelectric properties in LiNbO3 transferred layer by the Smart Cut™ technology for ultra wide band BAW filter applications. *Electron Devices Meeting, 2008. IEDM 2008.* 

[17] Hongyu Yu, Chuang-Yuan Lee, Wei Pang, Hao Zhang and Eun Sok Kim. Low Phase Noise, Low Power Consuming 3.7 GHz Oscillator Based on High-overtone Bulk

[18] M. Pijolat, A. Reinhardt, *et al.*. Large Qxf Product for HBAR using Smart Cut (TM) transfer of LiNbO3 thin layers onto LiNbO3 substrate. *IEEE Ultrasonics Symposium 1-4* 

[19] S. Ivanov, I. Koelyansky, G. Mansfeld and V. Veretin. Bulk Acoustic Wave High

[20] D. S. Bailey, M. M. Driscoll, and R. A. Jelen. Frequency Stability Of High Overtone Bulk

Acoustic Resonator. *2007 IEEE Ultrasonics Symposium*, pp.1160-1163, 2007

overtone resonator. *1990 1er congrès Français d'acoustique*, pp 599-601, 1990

Acous Tic Resonators. *1990 ultrasonics symposium.* pp. 509-512, 1990

resonators. *Proc. of the IEEE International Ultrasonics Symp.*, pp.201-204, 2008

[10] D. Royer & E. Dieulesaint. Elastic Waves in Solids II. *Springer*, 2000.

Microwave Sources. *1981 IEEE Ultrasonics Symposium*, pp.414-424, 1981

transfer. *Journal of Electronic Materials*, Vol. 30, n°7, 2001, pp. 834-840

Silicon. *IEEE Trans. on UFFC, Vol.54, n°4*, pp 870-876, 2007

*IEEE 1987 Ultrasonics Symposium.* pp. 501-504, 1987

All previous technological industrial development for solidly mounted resonator for instance could be easily use for fabrication of HBAR based on piezoelectric deposition. More technological development is required to control thickness, repeatability and so on (see section 4.1) for the second approach. In both case, packaging aspect is a key-point. Both side of HBAR need to be free for acoustic reason. Flip-chip approach seems to give the best result for industrial needs.

In both cases, design tool needs to be developed to realize conception of all HBAR devices. SAW design tool could be a good base to develop such software.

More works also need to improve performances or to fixe limits of all different HBAR devices which are specialized for each application. These developments are the precondition for industrial actions.

## **Author details**

T. Baron, E. Lebrasseur, F. Bassignot, G. Martin, V. Pétrini and S. Ballandras *FEMTO-ST, université de Franche-Comté, CNRS, ENSMM, UTBM, Département Temps-Fréquence, France* 

## **Acknowledgement**

This work was partly supported by the Centre National d'Etudes Spatiales (CNES) under grant #04/CNES/1941/00-DCT094 and still supported by the CNES under grant #R-S08/TC-0001-026, and by the Direction Generale pour l'Armement (DGA) under grant #05.34.016.

## **7. References**


[6] Curran Daniel R & Al, US Patent #3401275A, 1968-09-10

320 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

SAW design tool could be a good base to develop such software.

T. Baron, E. Lebrasseur, F. Bassignot, G. Martin, V. Pétrini and S. Ballandras

*FEMTO-ST, université de Franche-Comté, CNRS, ENSMM, UTBM, Département Temps-*

promote the industrial exploitation.

for industrial needs.

for industrial actions.

**Author details** 

*Fréquence, France* 

**7. References** 

2005

1993

*Symposium Proceedings*, pp. 834-837, 1980

**Acknowledgement** 

piezoelectric deposition gives easily high quality factor and frequency device. The largest potentiality of conception to address different applications is obtained by the second approach based on mono-crystal wafer assembled. Further developments required to

All previous technological industrial development for solidly mounted resonator for instance could be easily use for fabrication of HBAR based on piezoelectric deposition. More technological development is required to control thickness, repeatability and so on (see section 4.1) for the second approach. In both case, packaging aspect is a key-point. Both side of HBAR need to be free for acoustic reason. Flip-chip approach seems to give the best result

In both cases, design tool needs to be developed to realize conception of all HBAR devices.

More works also need to improve performances or to fixe limits of all different HBAR devices which are specialized for each application. These developments are the precondition

This work was partly supported by the Centre National d'Etudes Spatiales (CNES) under grant #04/CNES/1941/00-DCT094 and still supported by the CNES under grant #R-S08/TC-0001-026, and by the Direction Generale pour l'Armement (DGA) under grant #05.34.016.

[1] K. M. Lakin and J. S. Wang, UHF composite bulk wave resonators. *IEEE Ultrasonics* 

[2] K.M. Lakin, Thin film resonator technology. *IEEE Trans. on UFFC*, Vol.52, pp.707-716,

[3] K.M. Lakin, G.R. Kline, K.T. McCarron, High Q microwave acoustic resonators and filters. *IEEE Trans. on Microwave Theory and Techniques, Vol. 41, n°12,* pp,2139-2146, 1993 [4] S.P. Caldwell, M.M. Driscoll, D. Stansberry, D.S. Bailey, H.L. Salvo, High overtone bulk acoustic resonator frequency stability improvements. *IEEE Trans. on FCS,* pp. 744-748,

[5] D. Gachon, E. Courjon, G. Martin, L. Gauthier-Manuel, J.-C. Jeannot, W. Daniau and S. Ballandras, Fabrication of high frequency bulk acoustic wave resonator using thinned

single-crystal Lithium Niobate layers. *Ferroelectrics, Vol. 362*, pp. 30-40, 2008

	- [21] Lukas Baumgartel and Eun Sok Kim. Experimental Optimization of Electrodes for High Q, High Frequency HBAR. *2009 IEEE International Ultrasonics Symposium Proceedings.*  pp. 2107-2110, 2009

**Chapter 14** 

© 2013 Chatras et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2013 Chatras et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Modeling and Design of** 

**BAW Resonators and Filters for** 

**Integration in a UMTS Transmitter** 

Ji Fan, Dominique Cros, Michel Aubourg, Axel Flament,

**2. Model and design of bulk acoustic wave resonators** 

**2.1. Modeling of a BAW resonator in 1 dimension** 

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56026

bandwidth) up to few GHz [1].

the performances of these devices.

**1. Introduction** 

sections.

Matthieu Chatras , Stéphane Bila, Sylvain Giraud, Lise Catherinot,

Antoine Frappé, Bruno Stefanelli, Andreas Kaiser, Andreia Cathelin,

Jean Baptiste David, Alexandre Reinhardt, Laurent Leyssenne and Eric Kerhervé

Bulk-Acoustic Wave (BAW) resonators and filters are highly integrated devices, which represent an effective alternative for narrow-band components (up to 5% fractional

This chapter presents the integration of a BAW filter and of a BAW duplexer in a UMTS transmitter. The first section details one dimensional and three-dimensional techniques for the modeling and the design of BAW resonators. The second section proposes a synthesis approach for dimensioning BAW filters and the third section illustrates the approach with the characterization of several fabricated prototypes. Finally, The UMTS transmitter incorporating a BAW filter and a BAW duplexer is described with a particular emphasis on

The proposed method compares the impedance of a piezoelectric resonator obtained by both an electrical equivalent model and a piezoelectric model. By this way, it is possible to obtain the values of the electrical model as functions of all geometrical and material characteristics. The two models and their relation are described in the following

and reproduction in any medium, provided the original work is properly cited.

