**1. Introduction**

Since 70-ies, when the first delay lines and filters employing surface acoustic waves (SAW) were designed and fabricated, the use of SAW devices in special and commercial applications has expanded rapidly and the range of their working parameters was extended significantly (Hashimoto, 2000; Ruppel, 2001, 2002). In the last decade, their wide application in communication systems, cellular phones and base stations, wireless temperature and gas sensors has placed new requirements to SAW devices, such as very high operating frequencies (up to 10 GHz), low insertion loss, about 1 dB, high power durability, stable parameters at high temperatures etc.

The main element of a SAW device is a piezoelectric substrate with an interdigital transducer (IDT) used for generation and detection of SAW in the substrate. The number of single crystals utilized as substrates in SAW devices did not increase substantially since 70 ies because a new material must satisfy the list of strict requirements to be applied in commercial SAW devices: sufficiently strong piezoelectric effect, low or moderate variation of SAW velocity with temperature, low cost of as-grown large size crystals for fabrication of 4-inch wafers, long-term power durability, well developed and non-expensive fabrication process for SAW devices etc. Today only few single crystals are utilized as substrates in SAW devices: lithium niobate, LiNbO3 (LN), lithium tantalate, LiTaO3 (LT), quartz, SiO2, lithium tetraborate, Li2B4O7 (LBO), langasite, La3Ga5SiO14 (LGS) and some crystals of LGS group (LGT, LGN etc.) with similar properties.

The SAW velocities in these single crystals do not exceed 4000 m/s, which limit the highest operating frequencies of SAW devices by 2.5-3 GHz because of limitations imposed by the line-resolution technology of IDT fabrication. The minimum achievable insertion loss and maximum bandwidth of SAW devices depend on the electromechanical coupling coefficient, which can be evaluated for SAW as *k*2*≈*2*ΔV*/*V,* where Δ*V* is the difference between SAW velocities on free and electrically shorted surfaces. The largest values of *k2* can be obtained in some orientations of LN and LT. Ferroelectric properties of these materials are responsible for a strong piezoelectric effect. As a result, *k2* reaches 5.7% in LN and 1.2% in LT, for SAW. For leaky SAW (LSAW) propagating in rotated Y-cuts of both crystals, the coupling is higher and can exceed 20% for LN and 5% for LT. However, LSAW attenuates because of its leakage into the bulk waves when it propagates along the crystal surface. As a

Multilayered Structure as a Novel Material

for Surface Acoustic Wave Devices: Physical Insight 423

which includes all described examples, will be derived and a method of investigating this structure will be presented and then applied to few different structures, for which more

The first example of a layered structure is a dielectric isotropic silicon dioxide (SiO2) film deposited on one of rotated YX-cuts of LN or LT characterized by a high electromechanical coupling. Today these structures attract attention of researchers and SAW designers as the most promising candidates for application in SAW duplexers required in most of popular mobile phone systems (Kovacs et al., 2004; Kadota, 2007; Nakai et al., 2008; Nakanishi et al., 2008]. These RF devices must separate the transmitted and received signals in a narrow frequency interval and in a wide range of operating temperatures, e.g. between -30ºC and +85ºC. Therefore, substrate materials combining high propagation velocity, high electromechanical coupling and low TCF are strongly required. Due to the opposite signs of TCF in SiO2 and LT or LN, a layered SiO2/LT or SiO2/LN structure allows to obtain the desired combination of characteristics. When it is utilized in a resonator-type filter, the electrodes of IDT and metal gratings are commonly built at the interface between SiO2 and LN or LT. Besides, a heavy metal, such as copper, is utilized as an electrode material (Nakai et al., 2008). A layered structure with such electrode configuration is schematically presented as *Type 1* in Fig.1. The location of electrodes at the interface helps to keep a high electromechanical coupling and combine it with a large reflection coefficient in SAW resonators. A large metallization thickness effectively reduces resistive losses and results in a high *Q* factor of a SAW resonator. Another advantage of using heavy electrodes is a reduced propagation loss, which is achieved due to the transformation of LSAW into SAW. SAW devices with low insertion loss of 1-2 dB and low TCF of -7ppm/ºC, have been

A thickness of SiO2 film in SiO2/LT and SiO2/LN structures can vary within a wide range, from a few percent of a SAW wavelength up to a few wavelengths, to provide the required combination of electromechanical coupling, TCF and propagation loss. Moreover, SiO2 film helps to isolate the working surface of a SAW device from environmental influences and

*Type 1 Type 2* 

substrate substrate

Another electrode configuration in a structure with one thin film is schematically presented as *Type 2* in Fig. 1, with IDT located on the top surface. It is typical for a piezoelectric film on a non-piezoelectric substrate or on a substrate with low electromechanical coupling coefficient. As a piezoelectric film, zinc oxide ZnO is widely used (Kadota & Minakata, 1998; Nakahata et al., 2000; Emanetoglu et al., 2000; Brizoual et al., 2008). ZnO films are cheap and provide sufficiently high values of electromechanical coupling. The film deposition

film

IDT electrodes

detailed discussion of SAW propagation characteristics will be given.

successfully realized on such substrates (Kadota, 2007).

facilitates packaging of SAW chip.

film

Fig. 1. Two typical structures with one thin film

result, insertion loss of a SAW device increases. Attenuation coefficient depends on a crystal cut and IDT geometry. For example, in 36º to 48º rotated YX cuts of LT and in 41º to 76º YX rotated YX cuts of LN, high electromechanical coupling of LSAW can be combined with low attenuation coefficient via simultaneous optimization of orientation and electrode structure (Naumenko & Abbott, US patents, 2003, 2004). When these substrates are utilized in radiofrequency (RF) SAW filters with resonator-type structures, low insertion loss of 1dB or even less can be obtained. Today such low loss filters are widely used in mobile communication and navigation systems. The main drawback of these devices is high sensitivity of the characteristics to variations of temperature because the typical values of temperature coefficient of frequency (TCF) vary between -30 ppm/ºC and -40 ppm/ºC for LT and between -60 ppm/ºC and -75 ppm/ºC for LN.

Contrary to LN and LT, quartz is characterized by excellent temperature stability of SAW characteristics but low electromechanical coupling coefficient, *k2*<0.15%. Hence, even in resonator-type SAW filters with very narrow bandwidths, about 0.05%, where the loss of radiated energy is minimized due to the energy storage in a resonator, the best insertion loss achieved in a SAW device with matching circuits is only 2.5-4 dB.

In some orientations of LBO, LGS and other crystals of LGS group, zero TCF is combined with a moderate electromechanical coupling coefficient. However, these crystals have limited applications in commercial SAW devices because low SAW velocities restrict highfrequency applications on LGS and LBO dissolves in water and acid solutions, which prohibits application of conventional wafer fabrication processes to this material and finally results in an increased cost of SAW devices.

Hence, none of available single crystalline materials provides a combination of large piezoelectric coupling, zero TCF and high propagation velocity. A strong need in such material exists today, especially for application in SAW duplexers and multi-standard cellular phones, where the temperature compensation is the key issue because of necessity to divide a limited frequency bandwidth into few channels with no overlapping allowed in a wide range of operating temperatures. As an alternative to conventional SAW substrates, layered or multilayered (stratified) materials were studied extensively since 80-ies but only in the last decade some of these structures found commercial applications in SAW devices, due to the recent successes of thin film deposition technologies and development of robust simulation tools for design of SAW devices on layered structures.

### **2. Multilayered structures as materials for SAW devices**

As described above, the increasing requirements to the substrate materials, on one side, and rapid development of thin film deposition technologies, on the other side, gave rise to the novel class of materials for SAW devices – layered or multilayered structures. One or more films of different materials deposited on a regular substrate can improve its characteristics significantly. A proper combination of a substrate and a thin film helps to overcome the limitations of the conventional SAW substrates. In this section, a brief overview of the layered structures will be given. The examples presented here are currently investigated by different research groups as promising compound SAW substrate materials or found already some applications in SAW devices. The focus is made on the recent achievements in material research mostly motivated by challenges of the rapidly developing market of communication devices. Based on this overview, a generalized multilayered structure,

result, insertion loss of a SAW device increases. Attenuation coefficient depends on a crystal cut and IDT geometry. For example, in 36º to 48º rotated YX cuts of LT and in 41º to 76º YX rotated YX cuts of LN, high electromechanical coupling of LSAW can be combined with low attenuation coefficient via simultaneous optimization of orientation and electrode structure (Naumenko & Abbott, US patents, 2003, 2004). When these substrates are utilized in radiofrequency (RF) SAW filters with resonator-type structures, low insertion loss of 1dB or even less can be obtained. Today such low loss filters are widely used in mobile communication and navigation systems. The main drawback of these devices is high sensitivity of the characteristics to variations of temperature because the typical values of temperature coefficient of frequency (TCF) vary between -30 ppm/ºC and -40 ppm/ºC for

Contrary to LN and LT, quartz is characterized by excellent temperature stability of SAW characteristics but low electromechanical coupling coefficient, *k2*<0.15%. Hence, even in resonator-type SAW filters with very narrow bandwidths, about 0.05%, where the loss of radiated energy is minimized due to the energy storage in a resonator, the best insertion loss

In some orientations of LBO, LGS and other crystals of LGS group, zero TCF is combined with a moderate electromechanical coupling coefficient. However, these crystals have limited applications in commercial SAW devices because low SAW velocities restrict highfrequency applications on LGS and LBO dissolves in water and acid solutions, which prohibits application of conventional wafer fabrication processes to this material and finally

Hence, none of available single crystalline materials provides a combination of large piezoelectric coupling, zero TCF and high propagation velocity. A strong need in such material exists today, especially for application in SAW duplexers and multi-standard cellular phones, where the temperature compensation is the key issue because of necessity to divide a limited frequency bandwidth into few channels with no overlapping allowed in a wide range of operating temperatures. As an alternative to conventional SAW substrates, layered or multilayered (stratified) materials were studied extensively since 80-ies but only in the last decade some of these structures found commercial applications in SAW devices, due to the recent successes of thin film deposition technologies and development of robust

As described above, the increasing requirements to the substrate materials, on one side, and rapid development of thin film deposition technologies, on the other side, gave rise to the novel class of materials for SAW devices – layered or multilayered structures. One or more films of different materials deposited on a regular substrate can improve its characteristics significantly. A proper combination of a substrate and a thin film helps to overcome the limitations of the conventional SAW substrates. In this section, a brief overview of the layered structures will be given. The examples presented here are currently investigated by different research groups as promising compound SAW substrate materials or found already some applications in SAW devices. The focus is made on the recent achievements in material research mostly motivated by challenges of the rapidly developing market of communication devices. Based on this overview, a generalized multilayered structure,

LT and between -60 ppm/ºC and -75 ppm/ºC for LN.

results in an increased cost of SAW devices.

achieved in a SAW device with matching circuits is only 2.5-4 dB.

simulation tools for design of SAW devices on layered structures.

**2. Multilayered structures as materials for SAW devices** 

which includes all described examples, will be derived and a method of investigating this structure will be presented and then applied to few different structures, for which more detailed discussion of SAW propagation characteristics will be given.

The first example of a layered structure is a dielectric isotropic silicon dioxide (SiO2) film deposited on one of rotated YX-cuts of LN or LT characterized by a high electromechanical coupling. Today these structures attract attention of researchers and SAW designers as the most promising candidates for application in SAW duplexers required in most of popular mobile phone systems (Kovacs et al., 2004; Kadota, 2007; Nakai et al., 2008; Nakanishi et al., 2008]. These RF devices must separate the transmitted and received signals in a narrow frequency interval and in a wide range of operating temperatures, e.g. between -30ºC and +85ºC. Therefore, substrate materials combining high propagation velocity, high electromechanical coupling and low TCF are strongly required. Due to the opposite signs of TCF in SiO2 and LT or LN, a layered SiO2/LT or SiO2/LN structure allows to obtain the desired combination of characteristics. When it is utilized in a resonator-type filter, the electrodes of IDT and metal gratings are commonly built at the interface between SiO2 and LN or LT. Besides, a heavy metal, such as copper, is utilized as an electrode material (Nakai et al., 2008). A layered structure with such electrode configuration is schematically presented as *Type 1* in Fig.1. The location of electrodes at the interface helps to keep a high electromechanical coupling and combine it with a large reflection coefficient in SAW resonators. A large metallization thickness effectively reduces resistive losses and results in a high *Q* factor of a SAW resonator. Another advantage of using heavy electrodes is a reduced propagation loss, which is achieved due to the transformation of LSAW into SAW. SAW devices with low insertion loss of 1-2 dB and low TCF of -7ppm/ºC, have been successfully realized on such substrates (Kadota, 2007).

A thickness of SiO2 film in SiO2/LT and SiO2/LN structures can vary within a wide range, from a few percent of a SAW wavelength up to a few wavelengths, to provide the required combination of electromechanical coupling, TCF and propagation loss. Moreover, SiO2 film helps to isolate the working surface of a SAW device from environmental influences and facilitates packaging of SAW chip.

Fig. 1. Two typical structures with one thin film

Another electrode configuration in a structure with one thin film is schematically presented as *Type 2* in Fig. 1, with IDT located on the top surface. It is typical for a piezoelectric film on a non-piezoelectric substrate or on a substrate with low electromechanical coupling coefficient. As a piezoelectric film, zinc oxide ZnO is widely used (Kadota & Minakata, 1998; Nakahata et al., 2000; Emanetoglu et al., 2000; Brizoual et al., 2008). ZnO films are cheap and provide sufficiently high values of electromechanical coupling. The film deposition

Multilayered Structure as a Novel Material

velocity *V*≈8050 m/s, *k2* ≈1.42 %, TCF≈0.

2nd film

film

IDT

Substrate

1st

and low insertion loss was fabricated on this structure.

Fig. 2. Three typical structures with two thin films

combinations of film and substrate materials.

for Surface Acoustic Wave Devices: Physical Insight 425

resonator with center frequency about 2.5 GHz, temperature compensated characteristics

Nakahata (Nakahata et al., 1995) reported one more example of the two-layered structure, which can be referred to *Type 4.* It is a ZnO film on a silicon wafer with thin isotropic diamond layer between them. The following SAW characteristics have been obtained:

> 2nd film

 *Type 3 Type 4 Type 5* 

film

 2nd film

film

Substrate

1st

Substrate

The examples described above are not aimed at comprehensive survey of layered structures potentially applicable in SAW devices but demonstrate that a variety of layered structures can be referred to a few basic types. A unified approach to analysis of acoustic modes in different layered structures would be beneficial for optimization of SAW devices, because such approach allows comparing characteristics of the same SAW design built on different

The simulation of SAW characteristics is an important part of the SAW device design procedure. In a specified structure, such simulation must take into account orientation of each material if it is anisotropic, film thicknesses, a thickness and shape of IDT electrodes, electrode width to pitch ratio etc. Besides, the accurate analysis of all modes propagating in the investigated structure is required, including the main SAW or LSAW mode and all spurious modes generated by IDT in the specified frequency interval. A number of spurious modes grows with a number of layers and increasing of their thicknesses, which makes the simulation procedure more complicated. Moreover, with increasing film thickness SAW changes its nature and eventually transforms into a new type of acoustic wave. However, the characteristics of any acoustic mode change continuously with this transformation. The variation of film thicknesses within wide range helps to obtain a variety of *novel* materials with different combinations of characteristics demanded for SAW devices of different applications. After a proper combination of materials is selected, the geometrical parameters of a multilayered structure must be optimized to satisfy the desired electrical specification, including frequency bandwidth, insertion loss, out-of-band rejection, shape factor of frequency response or Q factor of a SAW resonator, temperature deviation of frequency etc. It is a common practice to optimize film and electrode thicknesses and other geometrical parameters of IDTs simultaneously with orientations of anisotropic materials

included in the layered structure, to achieve the best SAW device performance.

The challenges described above require a robust, fast and universal numerical technique, which could be applied to different types of multilayered structures, with film thicknesses varying within wide range and allowing transformation of SAW into boundary waves, plate modes or other types of acoustic waves. Such technique is described in the next section.

1st

technique (e.g. magnetron sputtering) has been well developed for this material. Another piezoelectric film, which is extensively studied as a promising material for high-frequency SAW devices is aluminum nitride (AlN) (Benetti et al., 2005, 2008; Fujii et al., 2008; Omori et al., 2008). It is characterized by chemical stability, mechanical strength, high acoustic velocity and good dielectric quality. Some other piezoelectric films, like CdS or GaN, were investigated previously but did not receive as much attention as ZnO or AlN.

A piezoelectric film is usually combined with silica glass, silicon, sapphire or diamond substrate. Silica glass is cheap, the use of silicon as a wafer enables simple integration of IF and RF components in one chip, sapphire is characterized by high SAW velocities, up to 6000 m/s, and diamond provides the highest SAW velocities among all materials, up to 11000 m/s, and is being used for high frequency SAW devices in the GHz range. For example, SAW resonator with center frequency about 4.5 GHz was built on AlN/diamond structure characterized by SAW velocity about 10000 m/s and *k2*≈1% (Omori et al., 2008). A combination of SAW velocity about 5500 m/s and electromechanical coupling about 0.25% can be obtained in AlN/sapphire structure (Ballandras et al., 2004). To reduce TCF of a SAW device, ZnO film can be combined with quartz or LGS. For example, nearly zero TCF and *k2* about 1.8% was achieved for SAW in ZnO/quartz structure, via optimization of quartz orientation (Kadota et al., 2008).

One more structure, which can be referred to the *Type 1*, recently found application in SAW devices. It is a thin plate of a piezoelectric crystal, such as LN or LT with thickness 10-15 wavelengths, which is directly bonded to a dielectric or semiconductor wafer. The bonding technology (Eda et al., 2000) provides excellent contact between the two materials and allows fabrication of SAW devices with reproducible characteristics on a thin LN or LT plate bonded to a thick silicon or glass wafer. In these structures, high values of electromechanical coupling coefficients typical for LN and LT are combined with improved TCFs, due to low thermal expansion coefficients (TCE) determined by massive silicon, glass or sapphire wafer (Tsutsumi et al., 2004). An example of bonded wafer will be numerically investigated in section 4.

The quality of a contact between LN or LT plate and a silicon wafer can be improved if a thin SiO2 film is deposited between these materials (Abbott et al., 2005). Such two-layered structure with IDT on the top surface is schematically shown as *Type 3* in Fig. 2. With silicon as a substrate, SiO2 as the first film and LN or LT as the second film (plate), this structure can give the same advantages as LT/Si or LN/Si bonded wafers but with higher quality contact between the materials. The presence of additional SiO2 film results in spurious acoustic modes propagating in a SAW device. These modes deteriorate the device performance and should be simulated properly to achieve the desired device characteristics. Another example of the *Type 3* structure is a silicon wafer with isotropic SiO2 as the first film and ZnO as the second film. Optimization of SiO2 and ZnO film thicknesses enables obtaining of a structure with TCF=0 (Emanetoglu et al., 2000). A high frequency SAW device can be built if SiO2 and ZnO films are deposited atop of a diamond or a sapphire substrate. With ZnO as the first film and isotropic SiO2 as the second film, the preferential location of IDT electrodes is at the substrate-film interface (*Type 4*). Alternatively, IDT can be built on ZnO surface and then buried in SiO2 overlay (*Type 5*). For example, Nakahata (Nakahata et al., 2000) reported on a SAW resonator using SiO2/ZnO/diamond structure with two different electrode configurations (*Type 4* and *Type 5*). Zero TCF, high velocity about 10000 m/s and *k2*≈1.2% were obtained for shear horizontally (SH) polarized SAW mode. A

technique (e.g. magnetron sputtering) has been well developed for this material. Another piezoelectric film, which is extensively studied as a promising material for high-frequency SAW devices is aluminum nitride (AlN) (Benetti et al., 2005, 2008; Fujii et al., 2008; Omori et al., 2008). It is characterized by chemical stability, mechanical strength, high acoustic velocity and good dielectric quality. Some other piezoelectric films, like CdS or GaN, were

A piezoelectric film is usually combined with silica glass, silicon, sapphire or diamond substrate. Silica glass is cheap, the use of silicon as a wafer enables simple integration of IF and RF components in one chip, sapphire is characterized by high SAW velocities, up to 6000 m/s, and diamond provides the highest SAW velocities among all materials, up to 11000 m/s, and is being used for high frequency SAW devices in the GHz range. For example, SAW resonator with center frequency about 4.5 GHz was built on AlN/diamond structure characterized by SAW velocity about 10000 m/s and *k2*≈1% (Omori et al., 2008). A combination of SAW velocity about 5500 m/s and electromechanical coupling about 0.25% can be obtained in AlN/sapphire structure (Ballandras et al., 2004). To reduce TCF of a SAW device, ZnO film can be combined with quartz or LGS. For example, nearly zero TCF and *k2* about 1.8% was achieved for SAW in ZnO/quartz structure, via optimization of quartz

One more structure, which can be referred to the *Type 1*, recently found application in SAW devices. It is a thin plate of a piezoelectric crystal, such as LN or LT with thickness 10-15 wavelengths, which is directly bonded to a dielectric or semiconductor wafer. The bonding technology (Eda et al., 2000) provides excellent contact between the two materials and allows fabrication of SAW devices with reproducible characteristics on a thin LN or LT plate bonded to a thick silicon or glass wafer. In these structures, high values of electromechanical coupling coefficients typical for LN and LT are combined with improved TCFs, due to low thermal expansion coefficients (TCE) determined by massive silicon, glass or sapphire wafer (Tsutsumi et al., 2004). An example of bonded wafer will be numerically investigated in

The quality of a contact between LN or LT plate and a silicon wafer can be improved if a thin SiO2 film is deposited between these materials (Abbott et al., 2005). Such two-layered structure with IDT on the top surface is schematically shown as *Type 3* in Fig. 2. With silicon as a substrate, SiO2 as the first film and LN or LT as the second film (plate), this structure can give the same advantages as LT/Si or LN/Si bonded wafers but with higher quality contact between the materials. The presence of additional SiO2 film results in spurious acoustic modes propagating in a SAW device. These modes deteriorate the device performance and should be simulated properly to achieve the desired device characteristics. Another example of the *Type 3* structure is a silicon wafer with isotropic SiO2 as the first film and ZnO as the second film. Optimization of SiO2 and ZnO film thicknesses enables obtaining of a structure with TCF=0 (Emanetoglu et al., 2000). A high frequency SAW device can be built if SiO2 and ZnO films are deposited atop of a diamond or a sapphire substrate. With ZnO as the first film and isotropic SiO2 as the second film, the preferential location of IDT electrodes is at the substrate-film interface (*Type 4*). Alternatively, IDT can be built on ZnO surface and then buried in SiO2 overlay (*Type 5*). For example, Nakahata (Nakahata et al., 2000) reported on a SAW resonator using SiO2/ZnO/diamond structure with two different electrode configurations (*Type 4* and *Type 5*). Zero TCF, high velocity about 10000 m/s and *k2*≈1.2% were obtained for shear horizontally (SH) polarized SAW mode. A

investigated previously but did not receive as much attention as ZnO or AlN.

orientation (Kadota et al., 2008).

section 4.

resonator with center frequency about 2.5 GHz, temperature compensated characteristics and low insertion loss was fabricated on this structure.

Nakahata (Nakahata et al., 1995) reported one more example of the two-layered structure, which can be referred to *Type 4.* It is a ZnO film on a silicon wafer with thin isotropic diamond layer between them. The following SAW characteristics have been obtained: velocity *V*≈8050 m/s, *k2* ≈1.42 %, TCF≈0.

Fig. 2. Three typical structures with two thin films

The examples described above are not aimed at comprehensive survey of layered structures potentially applicable in SAW devices but demonstrate that a variety of layered structures can be referred to a few basic types. A unified approach to analysis of acoustic modes in different layered structures would be beneficial for optimization of SAW devices, because such approach allows comparing characteristics of the same SAW design built on different combinations of film and substrate materials.

The simulation of SAW characteristics is an important part of the SAW device design procedure. In a specified structure, such simulation must take into account orientation of each material if it is anisotropic, film thicknesses, a thickness and shape of IDT electrodes, electrode width to pitch ratio etc. Besides, the accurate analysis of all modes propagating in the investigated structure is required, including the main SAW or LSAW mode and all spurious modes generated by IDT in the specified frequency interval. A number of spurious modes grows with a number of layers and increasing of their thicknesses, which makes the simulation procedure more complicated. Moreover, with increasing film thickness SAW changes its nature and eventually transforms into a new type of acoustic wave. However, the characteristics of any acoustic mode change continuously with this transformation.

The variation of film thicknesses within wide range helps to obtain a variety of *novel* materials with different combinations of characteristics demanded for SAW devices of different applications. After a proper combination of materials is selected, the geometrical parameters of a multilayered structure must be optimized to satisfy the desired electrical specification, including frequency bandwidth, insertion loss, out-of-band rejection, shape factor of frequency response or Q factor of a SAW resonator, temperature deviation of frequency etc. It is a common practice to optimize film and electrode thicknesses and other geometrical parameters of IDTs simultaneously with orientations of anisotropic materials included in the layered structure, to achieve the best SAW device performance.

The challenges described above require a robust, fast and universal numerical technique, which could be applied to different types of multilayered structures, with film thicknesses varying within wide range and allowing transformation of SAW into boundary waves, plate modes or other types of acoustic waves. Such technique is described in the next section.

Multilayered Structure as a Novel Material

Z

z=hm z=0

thickness *h* and dielectric permittivity *film*

recursive equation (Ingebrigtsen, 1969):

( ) ˆ

potential

( ) ˆ

φ

electrostatic potential

Fig. 3. Schematic drawing of analyzed multilayered structure

X

for Surface Acoustic Wave Devices: Physical Insight 427

Air or dielectric

p

Piezoelectric

Air or substrate

Analysis starts from the uppermost or lowermost half-infinite material, in which the wave structure is calculated. It can be a dielectric, a piezoelectric material or the air. In each adjacent finite-thickness layer, the transformation of the wave structure is deduced via separate treatment of incident and reflected partial modes. It means that the reflection and transmission matrix coefficients replace the transfer matrix to escape numerical noise at film thicknesses exceeding 3-5 wavelengths. For the structures with few dielectric (isotropic) films, the variation of the dielectric permittivity within each film characterized by the finite

ε

( )

*z th kh*

*film*

 ε

 ε

*z th kh*

( ) ( )

( ) ( )

 ε

*z h*

Milsom et al., 1977). This function relates the electric charge

ϕ, ε

*film*

*film*

where *k* is the wave number. Analysis of the lower and upper multilayered half-spaces is considered completed when the wave structure has been determined at *z*=0 and *z*=*h*m, where *h*m is a metal film thickness, and the surface impedance matrices ( ) ˆ

*Z k LOW* have been calculated at the upper and lower boundaries of the metal film. Each of these matrices characterizes the ratio between the vectors of displacements **u** and normal stresses **T** at the analyzed interface, ˆ <sup>1</sup> *Z* <sup>−</sup> = **uT** . A piezoelectric material is characterized by the generalized 4-dimensional displacement and stress vectors, with added electrostatic

*Z k LOW* comprise the information about the layers located above and below the metal film and enable simple formulation of electrical boundary conditions at *z*=0 and *z*=*h*m. If the mass load of metal film is included in the analysis of the upper *N* layers, then the function of *effective dielectric permittivity* (EDP) *ε<sup>s</sup>* ( ) *k* can be calculated at *z*=0 (Ingebrigtsen, 1969;

and normal electrical displacement *D*, respectively. The matrices ( ) ˆ

+ ⋅ + = + ⋅

ε

ε

Upper layers (n=1,…N)

Lower layers (m=1,…M)

is taken into account via the well known

σ

, (1)

*Z k UP* and

*Z k UP* and

at the surface to the

Electrodes or uniform metal film
