**4.1 SiO2/42ºYX LT with Al film at the interface**

The first example is a dielectric film on a piezoelectric substrate, which can be referred to the *Type 2* structure shown in Fig.1. The calculated characteristics of LSAWs propagating in SiO2/42ºYX LT with uniform Al film atop of the structure are presented in Fig. 4.

Fig. 4. Characteristics of leaky SAW propagating in SiO2/42ºYX LT with uniform Al film (*h*Al=5%λ) atop of SiO2 film, as functions of normalized SiO2 film thickness. OC or SC electrical conditions are analyzed

SiO2 is an isotropic dielectric film. LSAW velocities *V*, attenuation coefficients *δ* (in dB/*λ*, where *λ* is LSAW wavelength) and TCF are presented as functions of the normalized SiO2 film thickness, *h*/*λ*. These characteristics have been calculated for the open-circuited (OC) and short-circuited (SC) electrical conditions in Al film. The finite thickness of a metal film (*h*Al=5%*λ*) was taken into account. The difference between the OC and SC velocities determines the electromechanical coupling coefficient *k2*, which decreases rapidly with increasing dielectric film thickness. The behavior of attenuation coefficients depends on the electrical condition. The functions *δOC(h*SiO2) and *δSC(h*SiO2) reach nearly zero values at *h*SiO2=5%*λ* and *h*SiO2=8%*λ*, respectively. Therefore, the variation of SiO2 film thickness can be used for minimization of propagation losses in a SAW device. Due to the opposite signs of TCF in SiO2 film and LT substrate, in the layered structure the absolute value of TCF

Multilayered Structure as a Novel Material

Zno/sapphire etc. (Naumenko & Didenko, 1999).

**4.3 Al grating on 46ºYX LT/Si bonded wafer** 

Velocity [m/s]

4200

4100

4000

3900

3800

for Surface Acoustic Wave Devices: Physical Insight 431

Fig. 5, a shows the calculated velocities and attenuation coefficients of high-velocity LSAWs propagating in ZnO/sapphire structure, when the sapphire orientation is defined by the Euler angles (0°, -20.3°, 0°). The thickness of a metal film deposited atop of ZnO is not taken into account. The velocities of the fast quasi-shear and quasi-longitudinal LBAWs in sapphire are shown as VFAST SHEAR BAW and VLONGITUDINAL BAW, respectively. The nonattenuated SAW solution, which occurs on a high-velocity LSAW branch at *h*ZnO≈5.16%λ (Fig.5, a), is not a quasi-bulk wave described above. Fig.5,b shows the mechanical displacements, which follow the propagation of this wave. The analyzed solution is a sagittally polarized surface wave, which attenuates exponentially into the depth of sapphire, similar to Rayleigh SAW. Such *high velocity SAW* (HVSAW) can not exist without perturbation of a free crystal surface, e.g. by deposition of a thin film or a metal grating. The existence of this type of waves was revealed via numerical analysis of experimental data on SAW modes in ZnO/SiC structure (Didenko et al., 2000) and confirmed by other examples of layered structures, which support propagation of these waves, such as ZnO/diamond,

The HVSAW found in ZnO/sapphire may be attractive for applications in high-frequency SAW devices because it combines a high propagation velocity exceeding 9000 m/s with electromechanical coupling about 0.3%. With deposition of a metal film or a periodic metal grating the wave with attractive properties does not disappear but a combination of cut angle and ZnO thickness should be optimized properly to provide low LSAW attenuation.

In this example, SAW modes are investigated in LT/Si structure with a periodic Al grating atop of LT, which can be obtained experimentally by bonding LT plate to a silicon wafer. In

**Al** 

hLT

**46ºYX LT** 

**Silicon** 

such structure, the TCF may be dramatically reduced compared to regular LT wafer.

02468 Normalized LT thickness (wavelengths)

with OC and SC Al grating, as functions of normalized LT thickness, *h*Al=9%*λ*

Fig. 6. Velocities of acoustic modes propagating in 46ºYX LT plate bonded to silicon wafer,

VSC in regular LT

VOC in regular LT

SC OC

reduces with increasing film thickness but does not reach zero in the investigated interval of film thicknesses. However, larger SiO2 thicknesses are not considered because the electromechanical coupling coefficient becomes too low for practical applications. Further improvement of TCF is possible if IDT is located at the SiO2/LT interface (*Type 1* structure). Such example will be considered in Section 5.

### **4.2 ZnO/sapphire: Existence of high-velocity SAW**

The next example, ZnO film on a sapphire substrate, is a layered structure with a piezoelectric film on a non-piezoelectric substrate, which is potentially useful for highfrequency SAW device applications. Also this example demonstrates that a deposition of a thin film on a substrate can result in the existence of a high-velocity SAW, which cannot exist in a crystal without a thin film.

The typical SAW velocities in layered structures using a sapphire substrate are about 5500 m/s. Leaky SAWs, which have higher velocities, propagate with certain attenuation dependent on orientation of a substrate. Two types of LSAW can exist in crystals and layered structures: common or low-velocity LSAW, with velocities confined in the interval between that of the slow quasi-shear and fast quasi-shear *limiting* bulk acoustic waves (LBAWs), and high-velocity LSAW with velocities between that of the fast quasi-shear and quasi-longitudinal LBAWs. The *limiting* BAW is a bulk wave, which propagates in the sagittal plane (i.e. the plane, which is normal to the substrate surface and parallel to the propagation direction of SAW or LSAW) and is characterized by the group velocity parallel to the substrate surface. Usually a high velocity LSAW is not suitable for SAW device applications because of its fast attenuation. In some crystals with strong acoustic anisotropy, a high-velocity LSAW degenerates into the quasi-longitudinal LBAW in selected orientations, and low-attenuated LSAW can propagate around such orientations (Naumenko, 1996). For example, such waves exist in some orientations of quartz and LBO.

Fig. 5. (a) Velocity and attenuation coefficients of high-velocity leaky waves propagating in sapphire, Euler angles (0°, -20.3°, 0°), with ZnO film, as functions of normalized ZnO thickness, and (b) Displacements as function of depth, for HVSAW existing at *h*Zn0=5.16%λ

reduces with increasing film thickness but does not reach zero in the investigated interval of film thicknesses. However, larger SiO2 thicknesses are not considered because the electromechanical coupling coefficient becomes too low for practical applications. Further improvement of TCF is possible if IDT is located at the SiO2/LT interface (*Type 1* structure).

The next example, ZnO film on a sapphire substrate, is a layered structure with a piezoelectric film on a non-piezoelectric substrate, which is potentially useful for highfrequency SAW device applications. Also this example demonstrates that a deposition of a thin film on a substrate can result in the existence of a high-velocity SAW, which cannot

The typical SAW velocities in layered structures using a sapphire substrate are about 5500 m/s. Leaky SAWs, which have higher velocities, propagate with certain attenuation dependent on orientation of a substrate. Two types of LSAW can exist in crystals and layered structures: common or low-velocity LSAW, with velocities confined in the interval between that of the slow quasi-shear and fast quasi-shear *limiting* bulk acoustic waves (LBAWs), and high-velocity LSAW with velocities between that of the fast quasi-shear and quasi-longitudinal LBAWs. The *limiting* BAW is a bulk wave, which propagates in the sagittal plane (i.e. the plane, which is normal to the substrate surface and parallel to the propagation direction of SAW or LSAW) and is characterized by the group velocity parallel to the substrate surface. Usually a high velocity LSAW is not suitable for SAW device applications because of its fast attenuation. In some crystals with strong acoustic anisotropy, a high-velocity LSAW degenerates into the quasi-longitudinal LBAW in selected orientations, and low-attenuated LSAW can propagate around such orientations (Naumenko, 1996). For example, such waves exist in some orientations of quartz and LBO.

Fig. 5. (a) Velocity and attenuation coefficients of high-velocity leaky waves propagating in sapphire, Euler angles (0°, -20.3°, 0°), with ZnO film, as functions of normalized ZnO thickness, and (b) Displacements as function of depth, for HVSAW existing at *h*Zn0=5.16%λ

Such example will be considered in Section 5.

exist in a crystal without a thin film.

**4.2 ZnO/sapphire: Existence of high-velocity SAW** 

Fig. 5, a shows the calculated velocities and attenuation coefficients of high-velocity LSAWs propagating in ZnO/sapphire structure, when the sapphire orientation is defined by the Euler angles (0°, -20.3°, 0°). The thickness of a metal film deposited atop of ZnO is not taken into account. The velocities of the fast quasi-shear and quasi-longitudinal LBAWs in sapphire are shown as VFAST SHEAR BAW and VLONGITUDINAL BAW, respectively. The nonattenuated SAW solution, which occurs on a high-velocity LSAW branch at *h*ZnO≈5.16%λ (Fig.5, a), is not a quasi-bulk wave described above. Fig.5,b shows the mechanical displacements, which follow the propagation of this wave. The analyzed solution is a sagittally polarized surface wave, which attenuates exponentially into the depth of sapphire, similar to Rayleigh SAW. Such *high velocity SAW* (HVSAW) can not exist without perturbation of a free crystal surface, e.g. by deposition of a thin film or a metal grating. The existence of this type of waves was revealed via numerical analysis of experimental data on SAW modes in ZnO/SiC structure (Didenko et al., 2000) and confirmed by other examples of layered structures, which support propagation of these waves, such as ZnO/diamond, Zno/sapphire etc. (Naumenko & Didenko, 1999).

The HVSAW found in ZnO/sapphire may be attractive for applications in high-frequency SAW devices because it combines a high propagation velocity exceeding 9000 m/s with electromechanical coupling about 0.3%. With deposition of a metal film or a periodic metal grating the wave with attractive properties does not disappear but a combination of cut angle and ZnO thickness should be optimized properly to provide low LSAW attenuation.

## **4.3 Al grating on 46ºYX LT/Si bonded wafer**

In this example, SAW modes are investigated in LT/Si structure with a periodic Al grating atop of LT, which can be obtained experimentally by bonding LT plate to a silicon wafer. In such structure, the TCF may be dramatically reduced compared to regular LT wafer.

Fig. 6. Velocities of acoustic modes propagating in 46ºYX LT plate bonded to silicon wafer, with OC and SC Al grating, as functions of normalized LT thickness, *h*Al=9%*λ*

Multilayered Structure as a Novel Material

the admittance of a SAW resonator.

 SC OC

VOC in LT

VSC in LT

VOC in LT

VSC in LT

0 0.5 1 1.5 2 SiO2 thickness (wavelengths)

0 0.5 1 1.5 2 SiO2 thickness (wavelengths)

SAW devices.

3850

3850

*h*OVL=0.25%*λ*

3900

3950

4000

Velocity [m/s]

4050

4100

4150

3900

3950

4000

Velocity [m/s]

4050

4100

4150

for Surface Acoustic Wave Devices: Physical Insight 433

electrodes. Such layer is aimed at further improvement of the temperature characteristics of

In this case, Cu electrodes are investigated because this metal provides higher reflection coefficients in SAW resonators and hence lower insertion loss in RF SAW filters than Al, when electrodes are buried in SiO2 overlay. The velocities of acoustic modes are shown in Fig.8 as functions of the normalized intermediate SiO2 film thickness, while the thicknesses of SiO2 overlay, LT plate and Cu electrodes are fixed, *h*OVL=0.25%*λ*, *h*LT=4*λ* and *h*Cu=2.5%*λ*, respectively. The dispersion of SAW velocities in 46ºLT/SiO2/Si (Fig. 8,a) and SiO2/46ºLT/SiO2/Si (Fig. 8,b) structures demonstrates that in practice very accurate simulation is required to account for all spurious modes, because these modes may affect

**Cu** 

 **46ºYX LT** 

**4λ**

 **SiO2**

 **Silicon** 

 **SiO2** 

 **SiO2**

 **Silicon** 

 **46ºYX LT** 

**Cu** *h*OVL

(a)

(b)

Fig. 8. Velocities of acoustic modes as functions of normalized thickness of intermediate SiO2 film, when *h*LT=4*λ* and *h*Cu=2.5%*λ*, (a) in 46ºLT/SiO2/Si structure with Cu grating, and (b) in SiO2/46ºLT/SiO2/Si structure with Cu grating atop of 46ºLT plate and overlay thickness

The results shown in Fig.6 were obtained with the software SDA-FEM-SDA because the effect of a periodic metal grating is different from that of a uniform metal film. When LT thickness is about 1-2 wavelengths, the velocity of the SAW mode propagating in the layered structure is nearly the same as in a regular LT substrate with electrode thickness *h*Al=9%*λ* but the wave characteristics are perturbed by interactions with multiple plate modes, the number of which grows with increasing LT thickness. It should be noted that in a regular 46ºYX LT substrate the acoustic wave propagating with nearly the same velocity has a leaky wave nature. The bonding of LT plate to a silicon wafer results in the transformation of LSAW into SAW. The leakage of the wave becomes impossible because in silicon the shear BAW propagates faster than the analyzed SAW mode.

### **4.4 Al grating on 46ºYX LT/SiO2/Si bonded wafer**

The next example differs from the previous one by the additional SiO2 film between LT and Si wafer. A silicon dioxide layer is required to be deposited on the LT wafer to enable a stronger bond to silicon (Abbott et al., 2005). The presence of additional SiO2 film impacts the acoustic and electrical properties of a bonded wafer and SAW resonators built on its surface. The spectrum of acoustic modes propagating in LT/SiO2/Si bonded wafer looks more complicated than in LT/Si structure and depends on SiO2 and LT thicknesses. In Fig. 7, the velocities of acoustic modes are shown as functions of the normalized SiO2 film thickness when LT thickness is fixed, *h*LT=6λ.

Fig. 7. Velocities of acoustic modes propagating in 46ºYX LT/SiO2/Si structure with Al grating, as functions of normalized thickness of SiO2 film, when *h*LT=6*λ* and *h*Al=9%*λ*

### **4.5 Cu grating buried in SiO2, on 46ºYX LT/SiO2/Si bonded wafer**

This numerical example refers to the same bonded structure as described above but demonstrates the effect of additional SiO2 overlay deposited over periodic metal grating

The results shown in Fig.6 were obtained with the software SDA-FEM-SDA because the effect of a periodic metal grating is different from that of a uniform metal film. When LT thickness is about 1-2 wavelengths, the velocity of the SAW mode propagating in the layered structure is nearly the same as in a regular LT substrate with electrode thickness *h*Al=9%*λ* but the wave characteristics are perturbed by interactions with multiple plate modes, the number of which grows with increasing LT thickness. It should be noted that in a regular 46ºYX LT substrate the acoustic wave propagating with nearly the same velocity has a leaky wave nature. The bonding of LT plate to a silicon wafer results in the transformation of LSAW into SAW. The leakage of the wave becomes impossible because in

The next example differs from the previous one by the additional SiO2 film between LT and Si wafer. A silicon dioxide layer is required to be deposited on the LT wafer to enable a stronger bond to silicon (Abbott et al., 2005). The presence of additional SiO2 film impacts the acoustic and electrical properties of a bonded wafer and SAW resonators built on its surface. The spectrum of acoustic modes propagating in LT/SiO2/Si bonded wafer looks more complicated than in LT/Si structure and depends on SiO2 and LT thicknesses. In Fig. 7, the velocities of acoustic modes are shown as functions of the normalized SiO2 film

**Al** 

 **46ºYX LT** 

 **SiO2**

 **Silicon** 

silicon the shear BAW propagates faster than the analyzed SAW mode.

01234 SiO2 thickness (wavelengths)

**4.5 Cu grating buried in SiO2, on 46ºYX LT/SiO2/Si bonded wafer** 

hSiO2 VSC in regular LT

Fig. 7. Velocities of acoustic modes propagating in 46ºYX LT/SiO2/Si structure with Al grating, as functions of normalized thickness of SiO2 film, when *h*LT=6*λ* and *h*Al=9%*λ*

This numerical example refers to the same bonded structure as described above but demonstrates the effect of additional SiO2 overlay deposited over periodic metal grating

**4.4 Al grating on 46ºYX LT/SiO2/Si bonded wafer** 

thickness when LT thickness is fixed, *h*LT=6λ.

VOC in regular LT

Velocity [m/s]

4200

4100

4000

3900

3800

electrodes. Such layer is aimed at further improvement of the temperature characteristics of SAW devices.

In this case, Cu electrodes are investigated because this metal provides higher reflection coefficients in SAW resonators and hence lower insertion loss in RF SAW filters than Al, when electrodes are buried in SiO2 overlay. The velocities of acoustic modes are shown in Fig.8 as functions of the normalized intermediate SiO2 film thickness, while the thicknesses of SiO2 overlay, LT plate and Cu electrodes are fixed, *h*OVL=0.25%*λ*, *h*LT=4*λ* and *h*Cu=2.5%*λ*, respectively. The dispersion of SAW velocities in 46ºLT/SiO2/Si (Fig. 8,a) and SiO2/46ºLT/SiO2/Si (Fig. 8,b) structures demonstrates that in practice very accurate simulation is required to account for all spurious modes, because these modes may affect the admittance of a SAW resonator.

Fig. 8. Velocities of acoustic modes as functions of normalized thickness of intermediate SiO2 film, when *h*LT=4*λ* and *h*Cu=2.5%*λ*, (a) in 46ºLT/SiO2/Si structure with Cu grating, and (b) in SiO2/46ºLT/SiO2/Si structure with Cu grating atop of 46ºLT plate and overlay thickness *h*OVL=0.25%*λ*

Multilayered Structure as a Novel Material

propagating in isotropic structures, are listed below.

velocity is added between two half-infinite media.

ensure the propagation of a required acoustic wave.

improved temperature characteristics.

plate.

interfaces.

for Surface Acoustic Wave Devices: Physical Insight 435

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

a. In isotropic substrate with isotropic thin film, SH-polarized Love waves can propagate

b. Along the boundary between two rigidly connected isotropic half-infinite media, the sagittally polarized Stonely waves can propagate. These waves are usually trapped near

d. In isotropic thin plates, two types of waves can exist: sagittaly polarized Lamb waves (symmetric and anti-symmetric modes) and SH-polarized plate modes. With increasing

e. With plate thickness decreasing to zero, the first-order symmetric Lamb mode degenerates into the longitudinal BAW. With increasing plate thickness, this mode finally degenerates into two Rayleigh SAWs propagating along the boundaries of the

f. Higher-order plate modes arise from the shear and longitudinal BAWs at certain cut-off thicknesses and have a structure of standing waves propagating between two

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

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

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

plate thickness, higher-order modes appear and their number increases.

the interface between two media, with penetration depth about one wavelength. c. SH-polarized boundary waves can exist if additional thin film with lower shear BAW

if the shear BAW propagates faster in a substrate than in a film.

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

Fig. 9 demonstrates an example of calculated admittance of a periodic Cu grating used with different multilayered structures described above. In addition to the main SAW mode, which exhibits resonance and anti-resonance, the multiple spurious modes propagate in the analyzed layered structures and can disturb a SAW resonator performance. The frequencies of the spurious responses are very sensitive to the thicknesses of the layers in a multilayered structure. However, the accurate simulation of these modes enables optimization of the layered structure to minimize the effect of the spurious modes on a resonator performance.

Fig. 9. Admittance of infinite periodic grating with Al electrodes (*h*Al=10%*λ*), as function of normalized frequency, when the grating is built on regular 42ºYX LT substrate, on LT plate (*h*LT=5.5*λ*) bonded to Si substrate and atop of LT/SiO2/Si structure, with *h*SiO2=2*λ*. VBAW=4214.636 m/s

The effect of spurious modes on a resonator performance can be minimized by the variation of rotation angle of LT plate. If one of rotated YX cuts of LN is used as a piezoelectric plate, the insertion loss of a resonator SAW device can be reduced in a wider bandwidth.

The examples presented in this section illustrate possible applications of the developed numerical technique and demonstrate that being a part of design tools for SAW device simulation it can be also an efficient tool for analysis of acoustic modes in multilayered structures.

Fig. 9 demonstrates an example of calculated admittance of a periodic Cu grating used with different multilayered structures described above. In addition to the main SAW mode, which exhibits resonance and anti-resonance, the multiple spurious modes propagate in the analyzed layered structures and can disturb a SAW resonator performance. The frequencies of the spurious responses are very sensitive to the thicknesses of the layers in a multilayered structure. However, the accurate simulation of these modes enables optimization of the layered structure to minimize the effect of the spurious modes on a

LT

Anti-resonance

LT/Si

LT /SiO2/Si

0.45 0.46 0.47 0.48 0.49 0.5

Normalized frequency, fp/VBAW

Fig. 9. Admittance of infinite periodic grating with Al electrodes (*h*Al=10%*λ*), as function of normalized frequency, when the grating is built on regular 42ºYX LT substrate, on LT plate

The effect of spurious modes on a resonator performance can be minimized by the variation of rotation angle of LT plate. If one of rotated YX cuts of LN is used as a piezoelectric plate, the insertion loss of a resonator SAW device can be reduced in a wider

The examples presented in this section illustrate possible applications of the developed numerical technique and demonstrate that being a part of design tools for SAW device simulation it can be also an efficient tool for analysis of acoustic modes in multilayered

(*h*LT=5.5*λ*) bonded to Si substrate and atop of LT/SiO2/Si structure, with *h*SiO2=2*λ*.

resonator performance.

Normalized admittance

VBAW=4214.636 m/s

bandwidth.

structures.

1000

100

10

1

0.1

0.01

Resonance

0.001
