**3. An advanced numerical technique for analysis of acoustic waves in multilayered structures**

The most popular numerical technique used for simulation of SAW characteristic in multilayered structures is *Transfer Matrix Method* (TMM) (Adler, 1990). It is based on the matrix formalism suggested by Stroh (Stroh, 1965) for solution of a SAW problem in anisotropic media. For each material of a multilayered structure, TTM assumes building of a *fundamental acoustic tensor* dependent on the material constants and the analyzed orientation. Then the characteristics of the partial acoustic modes are found as the eigen vectors and eigen values of the matrix associated with this tensor. Finally, the *transfer matrix* is calculated, which characterizes the change of acoustic fields within the analyzed layer. The method is fast and convenient and does not impose any limitations on the number of layers. However, it is known to work unstable when the film thickness exceeds 3-5 wavelength because of bad conditioned matrices built of elements, some of which exponentially decay and others exponentially grow with film thickness. These elements are associated with *incident* and *reflected* modes. As suggested by Tan (Tan, 2002) and Reinhardt (Reinhardt et al., 2003), a separate treatment of these two groups of partial modes helps to avoid the instability and extends the range of the analyzed thicknesses from zero to infinite value.

Another limitation of the previously reported numerical techniques developed for analysis of SAW in multilayered structures is their focusing on a certain type of acoustic waves, which is related to a fixed type of a multilayered structure, with analytically defined boundary conditions at the interfaces and external boundaries. For example, acoustic waves propagating in a substrate with a thin film of finite thickness and *stress-free* boundary conditions at the top surface are different from acoustic waves propagating in the same combination of materials when the film thickness tends to infinite value. In practice, the results obtained with different versions of software using fixed boundary conditions often diverge and do not allow seeing how the wave characteristics change continuously with variation of film thickness within wide range. For example, the software *FEMSDA* (Endoh et al., 1995; Hashimoto et al., 2007, 2008), which is very popular among SAW device designers, includes separate versions for analysis of SAW in a substrate with a thin film and for investigation of boundary waves. In the first version, a film thickness providing robust calculations does not exceed a half-wavelength.

To overcome the described limitation, an advanced numerical technique was developed (Naumenko, 2009, 2010). It can be applied to a variety of multilayered structures and types of acoustic waves. The universal character of the software is achieved due to characterization of the air as a dielectric medium with a very small density and elastic stiffness constants and treatment of this medium as an example of a dielectric. The same numerical methods are applied to this and other materials, which compose a layered structure. With such approach, it is not necessary to fix the stress-free boundary conditions at the top surface of a thin film or at the boundaries of a finite-thickness plate to find acoustic waves propagating in these structures. The stress-free boundary conditions are automatically simulated with high accuracy when the adjacent medium is specified as the air.

The developed technique refers to a multilayered structure schematically shown in Fig. 3, in which a metal film or IDT is located between *N* upper and *M* lower layers, where *M* and *N* can vary between one and ten or more if necessary.

The most popular numerical technique used for simulation of SAW characteristic in multilayered structures is *Transfer Matrix Method* (TMM) (Adler, 1990). It is based on the matrix formalism suggested by Stroh (Stroh, 1965) for solution of a SAW problem in anisotropic media. For each material of a multilayered structure, TTM assumes building of a *fundamental acoustic tensor* dependent on the material constants and the analyzed orientation. Then the characteristics of the partial acoustic modes are found as the eigen vectors and eigen values of the matrix associated with this tensor. Finally, the *transfer matrix* is calculated, which characterizes the change of acoustic fields within the analyzed layer. The method is fast and convenient and does not impose any limitations on the number of layers. However, it is known to work unstable when the film thickness exceeds 3-5 wavelength because of bad conditioned matrices built of elements, some of which exponentially decay and others exponentially grow with film thickness. These elements are associated with *incident* and *reflected* modes. As suggested by Tan (Tan, 2002) and Reinhardt (Reinhardt et al., 2003), a separate treatment of these two groups of partial modes helps to avoid the instability and

Another limitation of the previously reported numerical techniques developed for analysis of SAW in multilayered structures is their focusing on a certain type of acoustic waves, which is related to a fixed type of a multilayered structure, with analytically defined boundary conditions at the interfaces and external boundaries. For example, acoustic waves propagating in a substrate with a thin film of finite thickness and *stress-free* boundary conditions at the top surface are different from acoustic waves propagating in the same combination of materials when the film thickness tends to infinite value. In practice, the results obtained with different versions of software using fixed boundary conditions often diverge and do not allow seeing how the wave characteristics change continuously with variation of film thickness within wide range. For example, the software *FEMSDA* (Endoh et al., 1995; Hashimoto et al., 2007, 2008), which is very popular among SAW device designers, includes separate versions for analysis of SAW in a substrate with a thin film and for investigation of boundary waves. In the first version, a film thickness providing robust

To overcome the described limitation, an advanced numerical technique was developed (Naumenko, 2009, 2010). It can be applied to a variety of multilayered structures and types of acoustic waves. The universal character of the software is achieved due to characterization of the air as a dielectric medium with a very small density and elastic stiffness constants and treatment of this medium as an example of a dielectric. The same numerical methods are applied to this and other materials, which compose a layered structure. With such approach, it is not necessary to fix the stress-free boundary conditions at the top surface of a thin film or at the boundaries of a finite-thickness plate to find acoustic waves propagating in these structures. The stress-free boundary conditions are automatically simulated with high accuracy when the adjacent medium is specified as the

The developed technique refers to a multilayered structure schematically shown in Fig. 3, in which a metal film or IDT is located between *N* upper and *M* lower layers, where *M* and *N*

**3. An advanced numerical technique for analysis of acoustic waves in** 

extends the range of the analyzed thicknesses from zero to infinite value.

calculations does not exceed a half-wavelength.

can vary between one and ten or more if necessary.

air.

**multilayered structures** 

Fig. 3. Schematic drawing of analyzed multilayered structure

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 thickness *h* and dielectric permittivity *film* ε is taken into account via the well known recursive equation (Ingebrigtsen, 1969):

$$\varepsilon\left(z+h\right) = \varepsilon\_{flm} \frac{\left[\varepsilon\left(z\right) + \varepsilon\_{flm} \cdot th\left(kh\right)\right]}{\left[\varepsilon\_{flm} + \varepsilon\left(z\right) \cdot th\left(kh\right)\right]}\tag{1}$$

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 UP* and ( ) 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 potential φ and normal electrical displacement *D*, respectively. The matrices ( ) ˆ *Z k UP* and ( ) ˆ *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; Milsom et al., 1977). This function relates the electric chargeσ at the surface to the electrostatic potential ϕ,

Multilayered Structure as a Novel Material

**4. Multilayered structures: Examples of analysis** 

VOC VSC

δSC

TCFSC

k2

structures of practical importance are presented.

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


electrical conditions are analyzed

0 0.001 0.002 0.003 0.004

TCF [ppm/oC]

Attenuation

[dB/wavelength]

Velocity [m/s]

for Surface Acoustic Wave Devices: Physical Insight 429

In this section, few examples of application of the developed numerical technique to the

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

δOC

Coupling [%]

 **Al**  SiO2

42ºYX LT

0 0.05 0.1 0.15 0.2

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

SiO2 thickness (wavelengths)

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

TCFOC

SiO2/42ºYX LT with uniform Al film atop of the structure are presented in Fig. 4.

$$
\sigma = |k| \cdot \varepsilon\_s(k) \cdot \varphi \tag{2}
$$

and can be used for extraction of the velocities and electromechanical coupling coefficients of SAW and other acoustic modes propagating in a multilayered structure. EDP function was originally introduced for semi-infinite piezoelectric medium, but it is also an efficient tool for analysis of acoustic waves in layered structures.

The numerical method described above was extended to a periodic metal grating sandwiched between two multilayered structures. In this case, the spectral domain analysis (SDA) of the upper and lower multi-layered half-spaces is combined with the finite-element method (FEM) applied to simulation of the electrode region. To some extent, the developed *SDA-FEM-SDA* technique (Naumenko, 2010) can be considered as an advanced FEMSDA method. In this case, the function of Harmonic Admittance *Y(f,s)* (Blotekjear et al, 1973; Zang et al, 1993) is calculated. Similar to EDP function, Harmonic Admittance relates the electric charge on the electrodes of the infinite periodic grating *Q* to the applied harmonically varying voltage *Ve* ,

$$Q = \left(j\omega\right)^{-1} Y(f\_\prime s) \cdot V\_\varepsilon \tag{3}$$

and depends on frequency *f* and the normalized wave number, *s p/* = *λ* , where *p* is a pitch of the grating. This function can be used for simulation of a SAW resonator and calculation of its main parameters: resonant and anti-resonant frequencies, reflection coefficient etc. Also it comprises the information about SAW and other acoustic modes, which can be generated in the analyzed layered structure, and their characteristics can be extracted from *Y(f,s)* .

It should be mentioned that a numerical procedure of finding eigen modes of the fundamental acoustic tensor can be successfully applied to the air as an example of isotropic medium and the calculated SAW characteristic do not differ noticeably from that obtained with stress-free conditions set analytically at the film/air interface. The method and software SDA-FEM-SDA enable analysis of electrodes composed of few different metal layers and having a complicated profile, with different edge angles in metal layers. The gaps between electrodes may be empty or filled with a dielectric material. Due to these options, some important physical effects can be simulated, such as the effect of a sublayer (e.g. titanium) often used for better adherence of electrode metal to the substrate or the effect of nonrectangular electrode profile on a SAW device performance.

The developed numerical technique can be applied to different types of multilayered structures and different acoustic waves can be investigated, for example,


In addition, a continuous transformation between different types of acoustic waves can be observed. It gives a physical insight into the mechanisms of wave transformation with increasing film thickness. An example of wave transformation will be considered in Section 5.

ϕ

(2)

= ⋅ (3)

*σ* =⋅ ⋅ *k ε*<sup>s</sup> ( ) *k*

tool for analysis of acoustic waves in layered structures.

harmonically varying voltage *Ve* ,

*Y(f,s)* .

Section 5.

and can be used for extraction of the velocities and electromechanical coupling coefficients of SAW and other acoustic modes propagating in a multilayered structure. EDP function was originally introduced for semi-infinite piezoelectric medium, but it is also an efficient

The numerical method described above was extended to a periodic metal grating sandwiched between two multilayered structures. In this case, the spectral domain analysis (SDA) of the upper and lower multi-layered half-spaces is combined with the finite-element method (FEM) applied to simulation of the electrode region. To some extent, the developed *SDA-FEM-SDA* technique (Naumenko, 2010) can be considered as an advanced FEMSDA method. In this case, the function of Harmonic Admittance *Y(f,s)* (Blotekjear et al, 1973; Zang et al, 1993) is calculated. Similar to EDP function, Harmonic Admittance relates the electric charge on the electrodes of the infinite periodic grating *Q* to the applied

> ( ) *<sup>1</sup> Q j<sup>ω</sup> Y( <sup>e</sup> f,s) V* <sup>−</sup>

and depends on frequency *f* and the normalized wave number, *s p/* = *λ* , where *p* is a pitch of the grating. This function can be used for simulation of a SAW resonator and calculation of its main parameters: resonant and anti-resonant frequencies, reflection coefficient etc. Also it comprises the information about SAW and other acoustic modes, which can be generated in the analyzed layered structure, and their characteristics can be extracted from

It should be mentioned that a numerical procedure of finding eigen modes of the fundamental acoustic tensor can be successfully applied to the air as an example of isotropic medium and the calculated SAW characteristic do not differ noticeably from that obtained with stress-free conditions set analytically at the film/air interface. The method and software SDA-FEM-SDA enable analysis of electrodes composed of few different metal layers and having a complicated profile, with different edge angles in metal layers. The gaps between electrodes may be empty or filled with a dielectric material. Due to these options, some important physical effects can be simulated, such as the effect of a sublayer (e.g. titanium) often used for better adherence of electrode metal to the substrate or the effect of

The developed numerical technique can be applied to different types of multilayered

In addition, a continuous transformation between different types of acoustic waves can be observed. It gives a physical insight into the mechanisms of wave transformation with increasing film thickness. An example of wave transformation will be considered in

nonrectangular electrode profile on a SAW device performance.

a. SAW and LSAW in a piezoelectric substrate;

structures and different acoustic waves can be investigated, for example,

b. SAW and LSAW in a substrate with one or few thin films (e.g. Love modes); c. plate modes generated by IDT and propagating in a thin plate (e.g. Lamb waves); d. boundary waves propagating along the interface between two half-infinite media; e. acoustic waves propagating in a thin piezoelectric plate bonded to a thick wafer.
