**5. Transformation of acoustic waves in anisotropic layered structures**

In isotropic or highly symmetric materials, acoustic waves are characterized by mechanical displacements either belonging to the sagittal plane, *u*=(*u*1, 0, *u*3), or normal to this plane, *u*=(0, *u*2, 0), i.e. such waves are Rayleigh-type or SH polarized modes. Similar solutions occur in some symmetric orientations of materials belonging to the lower symmetry classes. Such solutions have been extensively studied analytically. To the best author's knowledge, the most comprehensive overview of different types of acoustic waves existing in substrates with thin films, in thin plates and at the boundary between two half-infinite media was made by Viktorov (Viktorov, 1967, 1981). Some statements, which refer to acoustic waves propagating in isotropic structures, are listed below.


The layered structures used in SAW devices must include at least one anisotropic material to provide a piezoelectric coupling of SAW with IDT. Anisotropy results in mixed polarizations of acoustic modes propagating in thin films, plates and along the interface between two media. The transformation of each mode with increasing film thickness is unique and requires separate investigation. Whereas analytical study of such waves is possible only in some symmetric orientations, the numerical technique presented in this chapter enables calculation of the wave characteristics and analysis of displacements associated with different acoustic modes. Its application to multilayered structures can reveal the mechanisms of wave transformation. The understanding of these mechanisms helps to select properly the thicknesses of metal and dielectric or piezoelectric layers to ensure the propagation of a required acoustic wave.

An example of such investigation is presented here. It refers to 42ºYX LT with SiO2 film. Similar structure was considered in Section 4.1, but in the present example a periodic grating is analyzed instead of a thin metal film and this grating is located at the interface between LT substrate and SiO2 film. As a metal of the grating, copper is considered. Such structure is of great practical importance as potential material for RF SAW devices with improved temperature characteristics.

The calculated velocities of acoustic modes propagating in LT substrate with copper grating are shown in Fig.10,a as functions of the Cu electrode thickness. Fig.10,b shows the velocities

Multilayered Structure as a Novel Material

*p*=*λ*/2

(a)

(c)

(e)

(g)

*u***<sup>2</sup>**

for Surface Acoustic Wave Devices: Physical Insight 437

1.5λ

Fig. 11. Transformation of two SAW modes in 42ºYX LT with Cu grating with increasing

Fig.11 demonstrates how the acoustic fields associated with the modes SAW1 and SAW2 change with increasing SiO2 film thickness. OC electrical condition is considered, by way of example. The components of mechanical displacements in the sagittal plane are revealed as perturbations of the regular mesh and the values of SH components are presented as colored diagrams. When Cu thickness is small (Fig.11,a,b), SAW1 is nearly perfect Rayleigh wave and LSAW is a quasi-bulk SH-type wave slowly attenuating with depth. When *h*Cu>0.075*λ*, LSAW transforms into SAW2, which is also SH-type wave. At Cu thicknesses about 0.12*λ*, both SAW modes are perturbed by interaction between them. In the interval of

SiO2 film thickness; (a), (c), (e), (g) SAW1, (b), (d), (f), (h) SAW2

(b)

 

LT

(d)

0.2*λ*

 **Cu** SiO**<sup>2</sup>** LT

 *h*SiO2=1*λ* **Cu**  LT

SiO2

**Cu** 

LT

(f)

(h)

of acoustic waves propagating in SiO2/Cu grating/LT structure, as functions of the normalized SiO2 thickness, with Cu thickness fixed, *h*Cu=0.2λ. The analyzed Cu thicknesses look too high for application in SAW resonators but provide better insight into the wave transformation mechanisms, which is a purpose of this investigation.

When the metal thickness is small, two acoustic waves exist in LT, SAW1 and LSAW. With increasing Cu thickness, the velocities of both modes decrease rapidly and at *h*Cu=0.075*λ* LSAW transforms into the second SAW mode, SAW2. With further increasing of electrode thickness, two SAW modes interact with each other. To avoid discontinuities in the characteristics of two SAW modes, these modes are distinguished by their velocities: VSAW2>VSAW1.

Fig. 10. SAW and leaky SAW velocities in 42ºYX LT, (a) with Cu film, as functions of film thickness; (b) with Cu and SiO2 films, as functions of normalized SiO2 film thickness

If metal thickness is fixed (*h*Cu=0.2*λ*) and the gaps between electrodes are filled with SiO2, the velocities of two SAW modes grow rapidly (Fig.10,b). Another interaction between SAW1 and SAW2 occurs at *h*SiO2≈*h*Cu, i.e. when the top surface of the whole structure becomes flat. With increasing SiO2 film thickness two SAW modes finally transform into the boundary waves, BW1 and BW2. The boundary waves propagate with velocities lower than that of the shear BAW in SiO2. The wave BW2 shows electromechanical coupling sufficient for application in resonator SAW devices, *k2*=3.49%. For this mode, TCF grows from -31 ppm/ºC in LT substrate up to 10 ppm/ºC in a layered structure with SC grating and from - 43 ppm/ºC up to 6 ppm/ºC with OC grating. At *h*SiO2>0.7*λ*, higher-order plate modes arise from the fast shear limiting BAW in LT.

436 Acoustic Waves – From Microdevices to Helioseismology

of acoustic waves propagating in SiO2/Cu grating/LT structure, as functions of the normalized SiO2 thickness, with Cu thickness fixed, *h*Cu=0.2λ. The analyzed Cu thicknesses look too high for application in SAW resonators but provide better insight into the wave

When the metal thickness is small, two acoustic waves exist in LT, SAW1 and LSAW. With increasing Cu thickness, the velocities of both modes decrease rapidly and at *h*Cu=0.075*λ* LSAW transforms into the second SAW mode, SAW2. With further increasing of electrode thickness, two SAW modes interact with each other. To avoid discontinuities in the characteristics of two SAW modes, these modes are distinguished by their velocities:

VSBAW-SiO2

Fig. 10. SAW and leaky SAW velocities in 42ºYX LT, (a) with Cu film, as functions of film thickness; (b) with Cu and SiO2 films, as functions of normalized SiO2 film thickness

If metal thickness is fixed (*h*Cu=0.2*λ*) and the gaps between electrodes are filled with SiO2, the velocities of two SAW modes grow rapidly (Fig.10,b). Another interaction between SAW1 and SAW2 occurs at *h*SiO2≈*h*Cu, i.e. when the top surface of the whole structure becomes flat. With increasing SiO2 film thickness two SAW modes finally transform into the boundary waves, BW1 and BW2. The boundary waves propagate with velocities lower than that of the shear BAW in SiO2. The wave BW2 shows electromechanical coupling sufficient for application in resonator SAW devices, *k2*=3.49%. For this mode, TCF grows from -31 ppm/ºC in LT substrate up to 10 ppm/ºC in a layered structure with SC grating and from - 43 ppm/ºC up to 6 ppm/ºC with OC grating. At *h*SiO2>0.7*λ*, higher-order plate modes arise

0 12 3 hSi O2 (wavelengths)

 SiO2 **Cu**  LT

**(a) (b)** 

*h*SiO2

Boundary waves: BW1 BW2

*<sup>h</sup>*SiO2=*h*Cu High-order LSAW modes

1st 2nd

transformation mechanisms, which is a purpose of this investigation.

VSAW2>VSAW1.

5000

2000

0 0.1 0.2 hCU (wavelengths)

LSAW

OC

SC

SAW2

3000

VBAW1-LT

VBAW2-LT

SAW1

**Cu**  LT

from the fast shear limiting BAW in LT.

4000

Velocity [m/s]

Fig. 11. Transformation of two SAW modes in 42ºYX LT with Cu grating with increasing SiO2 film thickness; (a), (c), (e), (g) SAW1, (b), (d), (f), (h) SAW2

Fig.11 demonstrates how the acoustic fields associated with the modes SAW1 and SAW2 change with increasing SiO2 film thickness. OC electrical condition is considered, by way of example. The components of mechanical displacements in the sagittal plane are revealed as perturbations of the regular mesh and the values of SH components are presented as colored diagrams. When Cu thickness is small (Fig.11,a,b), SAW1 is nearly perfect Rayleigh wave and LSAW is a quasi-bulk SH-type wave slowly attenuating with depth. When *h*Cu>0.075*λ*, LSAW transforms into SAW2, which is also SH-type wave. At Cu thicknesses about 0.12*λ*, both SAW modes are perturbed by interaction between them. In the interval of

Multilayered Structure as a Novel Material

**6. Conclusion** 

one of the structures.

pp.7731-7741.

20, no. 12, pp. 1133–1138.

55, No 2, pp. 442–450.

309, San Juan, Puerto Rico, Sep. 2000.

**7. References** 

for Surface Acoustic Wave Devices: Physical Insight 439

multilayered structures and reveals another application of the numerical technique described in this chapter. Similar investigation can be performed for other structures of

In this chapter, some layered and multilayered structures, which look promising as substrates for modern SAW devices developed for applications in cellular phones, communication and navigation systems have been overviewed. A universal numerical technique, which enables fast and accurate analysis of these and other structures have been presented and, by way of example, applied to some multilayered structures of practical interest. The physical insight into the mechanisms of SAW transformation with increasing film thickness in a multilayered structure was provided via simulation of acoustic fields in

Abbott B.P., Naumenko N.F. & Caron J. (2005). Characterization of bonded wafer for RF

Adler E. L. (1990). Matrix methods applied to acoustic waves in multi-layers, *IEEE Trans.* 

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Blotekjear K., Ingebrigsten K. & Skeie H. (1973). A method for analyzing waves in structures

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practical importance or serve to study the wave processes in multilayered structures.

Cu thicknesses between 0.12*λ* and 0.2*λ*, the modes SAW1 and SAW2, which have been determined as the lower-velocity and higher velocity modes, exchange their polarizations. After the second interaction, which occurs at *h*SiO2=*h*Cu=0.2*λ*, SAW1 and SAW2 turn back into Rayleigh-type and SH-type waves, respectively. However, at *h*Cu=0.2*λ* (Fig.11,c,d) both waves still have mixed polarizations. With increasing SiO2 film thickness, SAW1 and SAW2 transform into the boundary waves BW1 and BW2, respectively (Fig. 11, g, h), with acoustic waves localized in Cu grating and around it. The boundary waves have mixed polarizations, which would be impossible in isotropic substrate with isotropic thin film, but due to specific features of the analyzed LT orientation, BW1 is nearly sagittally polarized wave and BW2 is nearly pure SH wave. BW2 penetrates deeper into SiO2 film than into LT substrate. A numerical analysis reveals that SiO2 thickness about 1.5*λ* is sufficient for transformation of SAW into the boundary wave.

Fig. 12. Acoustic fields associated with two higher-order modes propagating in 42ºYX LT with Cu grating and SiO2 film when *h*SiO2=3*λ*. (a) 1st mode; (b) 2nd mode

The acoustic fields associated with propagation of the two higher-order modes (Fig.10,b) have been also investigated. These modes have leaky wave nature. Fig.12 illustrates the structure of these modes at *h*SiO2=3*λ*. The first mode, which exists when *h*SiO2>0.7*λ*, has SH polarization deeply penetrating into SiO2 (Fig.12,a). With increasing SiO2 thickness, this mode degenerates into the SH BAW propagating in SiO2. The second mode, which exists at *h*SiO2>1.6*λ*, looks as a combination of SH-type SAW in LT substrate with Cu grating and sagittally polarized quasi-bulk wave propagating in SiO2 (Fig.12,b), with the amplitude of SH polarization component much higher than the amplitudes of two other components. This example demonstrates the effect of anisotropy on the propagation of acoustic waves in

  multilayered structures and reveals another application of the numerical technique described in this chapter. Similar investigation can be performed for other structures of practical importance or serve to study the wave processes in multilayered structures.
