**2.1 SAW parameters**

The most important parameter for SAW device design is the center frequency, which is determined by the period of the IDT fingers and the acoustic velocity. The governing equation that determines the operation frequency is:

SAW Parameters Analysis and Equivalent Circuit of SAW Device 445

2 22

∂ ∂∂ + =

*xx xx t* φ

*jk jk*

2 2 0 *<sup>l</sup>*

*xx xx*

**Position Mechanical conditions Electrical conditions** 

ε

*jk jk*

*ijkl kij*

*jkl jk*

*u*

*c e*

*e*

*l i*

*u u*

**x** 

x2

respectively, of the considered material.

The boundary conditions are shown in Table 1

1 3 3 *st <sup>S</sup> T T i i* <sup>=</sup>

*U U i i* = 1 2 3 3 *st nd T T i i* =

<sup>3</sup> <sup>0</sup> *nd*

U is the particle displacement.

is the scalar electric potential.

x3=0 1*st <sup>S</sup> U U i i* <sup>=</sup>

x3=h1 1 2 *st nd*

x3=h1 + h2 <sup>2</sup>

Table 1. Boundary conditions

Fig. 2. Multilayer structure

Constitutive equations:

where ,,, *ijkl ijk jk c e* ε ρ

φ

h1

h2

x3

substrate

2

Boundary is open

Boundary is open <sup>1</sup> <sup>2</sup> *st nd* φ

Boundary is short <sup>1</sup> <sup>2</sup> <sup>0</sup> *st nd*

 = , 1 2 *st nd D D*=

 φ

 φ= =

<sup>2</sup> 0 *nd* φ=

2 2 . . *nd nd D k* = −ε φ

Boundary is short

 = , <sup>1</sup>*st <sup>S</sup> D D*= Boundary is short <sup>1</sup> 0 *st <sup>S</sup>*

1*st S* φ φ

φ φ= =

φ

*<sup>T</sup> <sup>i</sup>* <sup>=</sup> Boundary is open

∂∂ ∂∂ ∂ (4)

∂ ∂ − = ∂∂ ∂∂ (5)

ρ

φ

are elastic tensor, piezoelectric tensor, dielectric tensor and mass density,

x1

1st layer

2nd layer

**… … …** 

$$\mathbf{f}\_0 = \mathbf{v}\_{\rm SAT} / \,\lambda\,\tag{1}$$

where

λ is the wavelength, determined by the periodicity of the IDT and vSAW is the acoustic wave velocity . For the technology being used in this research:

$$
\lambda = \mathbf{p} = \text{finger width} \times \mathbf{4} \tag{2}
$$

with the finger width (as shown in Figure 1) is determined by the design rule of the technology which sets the minimum metal to metal distance.

vSAW is surface acoustic wave velocity.

### Fig. 1. IDT parameters

By using matrix method or Finite Element Method (FEM) in section 2.2, the velocity v of acoustic wave is derived in two cases:



Therefore, the electromechanical coupling coefficient K is calculated approximately by Ingebrigtsen [54] as:

$$K^2 = 2\frac{V\_o - V\_s}{V\_o} \tag{3}$$

By using the matrix method or FEM and approximation of coupling factor as in (3), the SAW parameters in different structures AlN/Si, AlN/SiO2/Si and AlN/Mo/Si are calculated and analysed in three next sections.

### **2.2 Matrix method and Finite Element Method (FEM). The choice between them**

### **Matrix method**

The SAW propagation properties on one layer or multilayer structure are obtained by using matrix approach, proposed by J.J.Campbell and W.R.Jones [50], K.A.Ingebrigtsen [54], and then developed by Fahmy and Adler [31], [32], [33] and other authors [51], [52], [53]. The numerical solution method is based on characterizing each layer by means of a transfer matrix relating the mechanical and electrical field variables at the boundary planes. The boundary conditions for multilayer are based on the mechanical and electrical field variables those quantities that must be continuous at material interfaces. This matrix method is used to calculate the wave velocity and therefore, the electromechanical coupling factor. A general view and detail of this approach are given as follows and also presented in [50]-[53].

Fig. 2. Multilayer structure

Constitutive equations:

$$
\varepsilon\_{ijkl}\frac{\partial^2 u\_l}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} + e\_{kij}\frac{\partial^2 \phi}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} = \rho \frac{\partial^2 u\_i}{\partial t^2} \tag{4}
$$

$$
\varepsilon\_{jkl}\frac{\partial^2 u\_l}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} - \varepsilon\_{jk}\frac{\partial^2 \phi}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} = 0 \tag{5}
$$

where

444 Acoustic Waves – From Microdevices to Helioseismology

f0 = vSAW/ λ (1)

λ is the wavelength, determined by the periodicity of the IDT and vSAW is the acoustic wave

with the finger width (as shown in Figure 1) is determined by the design rule of the

Aperature (W)

By using matrix method or Finite Element Method (FEM) in section 2.2, the velocity v of

Therefore, the electromechanical coupling coefficient K is calculated approximately by

<sup>2</sup> 2 *o s o*

By using the matrix method or FEM and approximation of coupling factor as in (3), the SAW parameters in different structures AlN/Si, AlN/SiO2/Si and AlN/Mo/Si are calculated and

The SAW propagation properties on one layer or multilayer structure are obtained by using matrix approach, proposed by J.J.Campbell and W.R.Jones [50], K.A.Ingebrigtsen [54], and then developed by Fahmy and Adler [31], [32], [33] and other authors [51], [52], [53]. The numerical solution method is based on characterizing each layer by means of a transfer matrix relating the mechanical and electrical field variables at the boundary planes. The boundary conditions for multilayer are based on the mechanical and electrical field variables those quantities that must be continuous at material interfaces. This matrix method is used to calculate the wave velocity and therefore, the electromechanical coupling factor. A general view and detail of this approach are given as follows and also presented in [50]-[53].

**2.2 Matrix method and Finite Element Method (FEM). The choice between them** 

*V V <sup>K</sup> V*

width Finger

λ = p = finger width × 4 (2)

period (λ)

<sup>−</sup> <sup>=</sup> (3)

Finger

velocity . For the technology being used in this research:

technology which sets the minimum metal to metal distance.

vSAW is surface acoustic wave velocity.

acoustic wave is derived in two cases:



Fig. 1. IDT parameters

Ingebrigtsen [54] as:

**Matrix method** 

analysed in three next sections.

spacing

where

,,, *ijkl ijk jk c e* ε ρ are elastic tensor, piezoelectric tensor, dielectric tensor and mass density, respectively, of the considered material.

U is the particle displacement.

φis the scalar electric potential.

The boundary conditions are shown in Table 1


Table 1. Boundary conditions

SAW Parameters Analysis and Equivalent Circuit of SAW Device 447

In the design procedure of SAW devices, simple models like Equivalent Circuit Model coming from Smith Model and COM Model as presented above are used to achieve short calculation time and to get a general view of response of SAW devices. They are a good approach for designing SAW devices, for getting the frequency response, impedance parameters and transfer characteristics of SAW device. They could allow the designer to determine the major dimensions and parameters in number of fingers, finger width, and

Field theory is the most appropriate theory for the design SAW devices as it involves the resolution of all the partial differential equations for a given excitation. The Finite Element Model (FEM) is the most appropriate numerical representation of field theory where the piezoelectric behaviour of the SAW devices can be discretized [45], [46]. Besides, nowadays, FEM tools also provide 3D view for SAW device, such as COMSOL® [47], Coventor® [48],

The typical SAW devices can include a lot of electrodes (hundreds or even thousands of electrodes). In fact, we would like to include as many IDT finger pairs as possible in our FEM simulations. This would however significantly increase the scale of the device. Typically finite element models of SAW devices require a minimum of 20 mesh elements per wavelength to ensure proper convergence. A conventional two-port SAW devices consisting of interdigital transducers (IDT) may have – especially on substrate materials with low piezoelectric coupling constants - a length of thousands of wavelengths and an aperture of hundred wavelengths. Depending on the working frequency, the substrate which carries the electrode also has a depth of up to one hundred wavelengths. Taking into account that FEM requires a spatial discretization with at least twenty first order finite elements per wavelength and that an arbitrary piezoelectric material has at least four degrees of freedom, this leads to 8 x 108 unknowns in the three dimensional (3-D) case. Hence, the 3-D FEM representation of SAW device with hundreds of IDT fingers would require several million elements and nodes. The computational cost to simulate such a device is extremely high, or the

Fortunately, SAW devices consist of periodic section. M.Hofer et al proposed the Periodic Boundary Condition (PBC) in the FEM that allows the reduction of size of FE model

A good agreement between FEM and analytic method is obtained via the results in case of

From this table, matrix method and FEM give the same results. However, FEM would takes a long time and require a trial and error to find the results. Consequently, to reduce time, the matrix method proposed to be used to extract the parameters of SAW devices; FEM is

Matrix method FEM Difference between Matrix

method and FEM (%)

aperture. However, they are subjected to some simplifications and restrictions.

amount of elements could not be handled by nowadays computer resources.

SAW with AlN thickness of 4μm, wavelength of 8μm presented in Table 2.

f0 (MHz) 771.13 775.48 0.56 fs (MHz) 770.26 774.57 0.56 v0 (m/s) 6169.02 6203.87 0.56 vs (m/s) 6162.07 6196.54 0.56 K (%) 4.74 4.86 2.4

Table 2. Comparison between matrix method and FEM

**Finite Element Method (FEM)** 

ANSYS® [49].

tremendously [45], [46].

where D: electronic displacement,

$$D\_k = \frac{\partial \phi}{\partial \mathbf{x}\_k} \tag{6}$$

The general solution for Ul and φ(1) and (2) may be written as follows:

$$\mathcal{U}I\_l = \sum\_{m=1}^{n} \mathbb{C}\_m A\_l^{(m)} \exp[ik(b^{(m)}\mathbf{x}\_3 + \mathbf{x}\_1 - vt)] \tag{7}$$

where l =1, 2, 3

$$\phi = \sum\_{m=1}^{n} \mathbb{C}\_{m} A\_{4}^{\ \ \ \left(m\right)} \exp\left[ik\left(b^{\left(m\right)}\mathbf{x}\_{3} + \mathbf{x}\_{1} - vt\right)\right] \tag{8}$$

The coefficients Cm are determined from boundary conditions.

By substituting (7) and (8) in every layer into the boundary conditions, we have general form

$$\begin{bmatrix} \mathbf{C}\_1^S \\ \cdots \\ \mathbf{C}\_4^S \\ \mathbf{C}\_4^{\*t} \\ \vdots \\ \mathbf{C}\_8^{\*t} \\ \mathbf{C}\_8^{\*u} \\ \cdots \\ \cdots \\ \mathbf{C}\_8^{\*u'} \end{bmatrix} = \mathbf{0} \tag{9}$$

Phase velocity is determined from the condition:

$$\text{Det}(\text{H}) = 0 \tag{10}$$

(use approximation to solve (10))

Figure 3 shows the wave velocity of structure AlN/SiO2(1.3μm)/Si(4μm).

Fig. 3. Wave velocity in structure AlN/SiO2(1.3μm)/Si(4μm) with different thicknesses of AlN

### **Finite Element Method (FEM)**

446 Acoustic Waves – From Microdevices to Helioseismology

*k*

(1) and (2) may be written as follows:

exp[ ( )]

exp[ ( )]

3 1

= + <sup>−</sup> (7)

= + <sup>−</sup> (8)

<sup>=</sup> ∂ (6)

(9)

Det(H)=0 (10)

*x* ∂φ

( ) ( )

( ) ( ) 4 3 1

By substituting (7) and (8) in every layer into the boundary conditions, we have general

1

*S*

...

 <sup>=</sup> 

*C*

*C C*

4 1 1

*S*

*st*

*C C* *st nd*

... 0

2 8

Fig. 3. Wave velocity in structure AlN/SiO2(1.3μm)/Si(4μm) with different thicknesses of AlN

*nd*

...

*C*

*C A ik b x x vt*

*U C A ik b x x vt*

*k*

*D*

*<sup>n</sup> m m*

*<sup>n</sup> m m*

[ ]

Figure 3 shows the wave velocity of structure AlN/SiO2(1.3μm)/Si(4μm).

*H*

φ

1

=

1

=

The coefficients Cm are determined from boundary conditions.

Phase velocity is determined from the condition:

(use approximation to solve (10))

*m m* φ

*l ml m*

where D: electronic displacement,

The general solution for Ul and

where l =1, 2, 3

form

In the design procedure of SAW devices, simple models like Equivalent Circuit Model coming from Smith Model and COM Model as presented above are used to achieve short calculation time and to get a general view of response of SAW devices. They are a good approach for designing SAW devices, for getting the frequency response, impedance parameters and transfer characteristics of SAW device. They could allow the designer to determine the major dimensions and parameters in number of fingers, finger width, and aperture. However, they are subjected to some simplifications and restrictions.

Field theory is the most appropriate theory for the design SAW devices as it involves the resolution of all the partial differential equations for a given excitation. The Finite Element Model (FEM) is the most appropriate numerical representation of field theory where the piezoelectric behaviour of the SAW devices can be discretized [45], [46]. Besides, nowadays, FEM tools also provide 3D view for SAW device, such as COMSOL® [47], Coventor® [48], ANSYS® [49].

The typical SAW devices can include a lot of electrodes (hundreds or even thousands of electrodes). In fact, we would like to include as many IDT finger pairs as possible in our FEM simulations. This would however significantly increase the scale of the device. Typically finite element models of SAW devices require a minimum of 20 mesh elements per wavelength to ensure proper convergence. A conventional two-port SAW devices consisting of interdigital transducers (IDT) may have – especially on substrate materials with low piezoelectric coupling constants - a length of thousands of wavelengths and an aperture of hundred wavelengths. Depending on the working frequency, the substrate which carries the electrode also has a depth of up to one hundred wavelengths. Taking into account that FEM requires a spatial discretization with at least twenty first order finite elements per wavelength and that an arbitrary piezoelectric material has at least four degrees of freedom, this leads to 8 x 108 unknowns in the three dimensional (3-D) case. Hence, the 3-D FEM representation of SAW device with hundreds of IDT fingers would require several million elements and nodes. The computational cost to simulate such a device is extremely high, or the amount of elements could not be handled by nowadays computer resources.

Fortunately, SAW devices consist of periodic section. M.Hofer et al proposed the Periodic Boundary Condition (PBC) in the FEM that allows the reduction of size of FE model tremendously [45], [46].

A good agreement between FEM and analytic method is obtained via the results in case of SAW with AlN thickness of 4μm, wavelength of 8μm presented in Table 2.


Table 2. Comparison between matrix method and FEM

From this table, matrix method and FEM give the same results. However, FEM would takes a long time and require a trial and error to find the results. Consequently, to reduce time, the matrix method proposed to be used to extract the parameters of SAW devices; FEM is

SAW Parameters Analysis and Equivalent Circuit of SAW Device 449

The coupling factor K for this kind of device is shown in Figure 5. When normalized thickness of AlN layer is larger than 3, the coupling factor K still remain at 4.74% by that the

Wave velocity and coupling factor in structure AlN/SiO2/Si are also presented in Figure 6

In this configuration, K is at its maximum value of 5.34% when khAlN=0.55.

Fig. 6. Dependence of wave velocity in SAW device AlN/SiO2/Si substrate on the

Fig. 7. Dependence of coupling factor K(%) in SAW device AlN/ SiO2/Si substrate on the

In this configuration, as results in Figure 6, when 6 *khAlN* < , with the same thickness of AlN layer, an increase in thickness of SiO2 would decrease the wave velocity. When *khAlN* > 6 , the wave velocity reaches the velocity of the Rayleigh wave in AlN substrate v(AlN substrate)=6169 (m/s). A same conclusion is formulated also for coupling factor for this kind of structure, AlN/SiO2/Si, in Figure 7; when 6 *khAlN* > , K remains at the value of 4.7%.

normalized thickness khAlN of AlN layer and khSiO2

normalized thickness khAlN of AlN layer and khSiO2

**2.4 Wave velocity, coupling factor in AlN/SiO2/Si structure** 

wave travels principally in AlN layer.

and Figure 7, respectively.

used to get a 3D view and explain some results that can not be explained by equivalent circuit. This point will be presented in next sections.

The three next sections present and analyse SAW parameters in different structures AlN/Si, AlN/SiO2/Si and AlN/Mo/Si.

### **2.3 Wave velocity, coupling factor in AlN/Si structure**

Figure 4 shows the dependence of Rayleigh wave velocity V0 and Vs on the normalized thickness as respect to the wavelength, khAlN of AlN layer in SAW device AlN/Si substrate, where normalized thickness is defined by:

$$kh = \frac{2\pi h}{\lambda} \tag{11}$$

In this graph, when the normalized thickness of AlN, khAlN is larger than 3, the wave velocity reaches the velocity of the Rayleigh wave in AlN substrate v(AlN substrate)=6169 (m/s). This could be explained that the wave travels principally in AlN layer when khAlN is larger than 3, because for low frequency the wave penetrates inside the other layer and this work is in the case where the wave are dispersive. It is better to be in the frequency range where the Rayleigh wave is obtained to have a constant velocity.

Fig. 4. Calculated values of wave velocity V0 and Vs in SAW device AlN/Si substrate depend on the normalized thickness khAlN of AlN layer

Fig. 5. Calculated values of coupling factor K(%) in SAW device AlN/Si substrate depends on the normalized thickness khAlN of AlN layer

used to get a 3D view and explain some results that can not be explained by equivalent

The three next sections present and analyse SAW parameters in different structures AlN/Si,

Figure 4 shows the dependence of Rayleigh wave velocity V0 and Vs on the normalized thickness as respect to the wavelength, khAlN of AlN layer in SAW device AlN/Si

> <sup>2</sup> *<sup>h</sup> kh* π

In this graph, when the normalized thickness of AlN, khAlN is larger than 3, the wave velocity reaches the velocity of the Rayleigh wave in AlN substrate v(AlN substrate)=6169 (m/s). This could be explained that the wave travels principally in AlN layer when khAlN is larger than 3, because for low frequency the wave penetrates inside the other layer and this work is in the case where the wave are dispersive. It is better to be in the frequency range where the

Fig. 4. Calculated values of wave velocity V0 and Vs in SAW device AlN/Si substrate

Fig. 5. Calculated values of coupling factor K(%) in SAW device AlN/Si substrate depends

λ

<sup>=</sup> (11)

circuit. This point will be presented in next sections.

**2.3 Wave velocity, coupling factor in AlN/Si structure** 

substrate, where normalized thickness is defined by:

Rayleigh wave is obtained to have a constant velocity.

depend on the normalized thickness khAlN of AlN layer

on the normalized thickness khAlN of AlN layer

AlN/SiO2/Si and AlN/Mo/Si.

The coupling factor K for this kind of device is shown in Figure 5. When normalized thickness of AlN layer is larger than 3, the coupling factor K still remain at 4.74% by that the wave travels principally in AlN layer.

In this configuration, K is at its maximum value of 5.34% when khAlN=0.55.

### **2.4 Wave velocity, coupling factor in AlN/SiO2/Si structure**

Wave velocity and coupling factor in structure AlN/SiO2/Si are also presented in Figure 6 and Figure 7, respectively.

Fig. 6. Dependence of wave velocity in SAW device AlN/SiO2/Si substrate on the normalized thickness khAlN of AlN layer and khSiO2

Fig. 7. Dependence of coupling factor K(%) in SAW device AlN/ SiO2/Si substrate on the normalized thickness khAlN of AlN layer and khSiO2

In this configuration, as results in Figure 6, when 6 *khAlN* < , with the same thickness of AlN layer, an increase in thickness of SiO2 would decrease the wave velocity. When *khAlN* > 6 , the wave velocity reaches the velocity of the Rayleigh wave in AlN substrate v(AlN substrate)=6169 (m/s). A same conclusion is formulated also for coupling factor for this kind of structure, AlN/SiO2/Si, in Figure 7; when 6 *khAlN* > , K remains at the value of 4.7%.

SAW Parameters Analysis and Equivalent Circuit of SAW Device 451

wave velocity and coupling factor K. These influences are shown in Figure 9 and Figure 10,

Fig. 10. Coupling factor K(%) in SAW device AlN/Mo/Si substrate depends on the

**1.02** 

From Figure 9, the use of Mo layer would increase the wave velocity with any thickness of AlN layer and Mo layer. In case of coupling factor K as in Figure 10, the Mo layer, however, could decrease K when the khAlN is less than 1.02. When the normalized thickness of AlN layer khAlN is in the range from 1.17 to 2.7, the Mo layer would increase the coupling factor K. And when the khAlN is larger than 2.7, the Mo has no influence on wave velocity and coupling factor. The reason of this effect could be explained by the displacement profile in AlN/Mo/Si structure, as shown in Figure 11 for thickness AlN value of khAlN=2.7. We could note that when 2.7 *khAlN* ≥ , the first interesting point is that the wave travels principally in AlN layer and Si substrate, the second one is that the relative displacement U/Umax in Mo layer will be smaller than 0.5. These points would explain the reason why when 2.7 *khAlN* ≥ the use of Mo has no influence on wave velocity and coupling factor.

Fig. 11. Displacement profile along the depth of the multilayer AlN/Mo/Si, khAlN=2.7

normalized thickness khAlN and khMo

respectively.

To understand the above behavior, we use FEM method to display displacement profile along the depth of multilayer AlN/SiO2/Si. These results obtained from FEM method in case of khSiO2=0.7854, khAlN=5 and khAlN=0.2 are compared as in Figure 8.

Fig. 8. Displacement profile along the depth of the multilayer AlN/SiO2/Si, khSiO2=0.7854

From Figure 8, we note that wave travels principally in AlN layer for a khAlN value of 5. By this reason, from a khAlN value of larger than 5, the coupling factor K remains at 4.7% and wave velocity remains at 6169m/s. For khAlN=0.2, wave travels principally in SiO2 layer and Si substrate that are not piezoelectric layer. Consequently, the coupling factor K reaches the value of 0%.

In conclusion, the values of wave velocity and coupling factor depend on wave propagation medium, in which constant values of wave velocity and coupling factor indicate a large contribution of AlN layer, and coupling factor value of near 0% indicates a large contribution of SiO2 layer and Si substrate.

### **2.5 Wave velocity, coupling factor in AlN/Mo/Si structure**

For our devices, a thin Mo layer will be also deposited below the AlN layer to impose the crystal orientation of AlN. Besides this dependence, the Mo layer also has influences on

Fig. 9. Wave velocity AlN/Mo/Si substrate depends on the normalized thickness khAlN and khMo

To understand the above behavior, we use FEM method to display displacement profile along the depth of multilayer AlN/SiO2/Si. These results obtained from FEM method in

Fig. 8. Displacement profile along the depth of the multilayer AlN/SiO2/Si, khSiO2=0.7854 From Figure 8, we note that wave travels principally in AlN layer for a khAlN value of 5. By this reason, from a khAlN value of larger than 5, the coupling factor K remains at 4.7% and wave velocity remains at 6169m/s. For khAlN=0.2, wave travels principally in SiO2 layer and Si substrate that are not piezoelectric layer. Consequently, the coupling factor K reaches

In conclusion, the values of wave velocity and coupling factor depend on wave propagation medium, in which constant values of wave velocity and coupling factor indicate a large contribution of AlN layer, and coupling factor value of near 0% indicates a large

For our devices, a thin Mo layer will be also deposited below the AlN layer to impose the crystal orientation of AlN. Besides this dependence, the Mo layer also has influences on

Fig. 9. Wave velocity AlN/Mo/Si substrate depends on the normalized thickness khAlN

the value of 0%.

and khMo

contribution of SiO2 layer and Si substrate.

**2.5 Wave velocity, coupling factor in AlN/Mo/Si structure** 

case of khSiO2=0.7854, khAlN=5 and khAlN=0.2 are compared as in Figure 8.

wave velocity and coupling factor K. These influences are shown in Figure 9 and Figure 10, respectively.

Fig. 10. Coupling factor K(%) in SAW device AlN/Mo/Si substrate depends on the normalized thickness khAlN and khMo

From Figure 9, the use of Mo layer would increase the wave velocity with any thickness of AlN layer and Mo layer. In case of coupling factor K as in Figure 10, the Mo layer, however, could decrease K when the khAlN is less than 1.02. When the normalized thickness of AlN layer khAlN is in the range from 1.17 to 2.7, the Mo layer would increase the coupling factor K. And when the khAlN is larger than 2.7, the Mo has no influence on wave velocity and coupling factor. The reason of this effect could be explained by the displacement profile in AlN/Mo/Si structure, as shown in Figure 11 for thickness AlN value of khAlN=2.7. We could note that when 2.7 *khAlN* ≥ , the first interesting point is that the wave travels principally in AlN layer and Si substrate, the second one is that the relative displacement U/Umax in Mo layer will be smaller than 0.5. These points would explain the reason why when 2.7 *khAlN* ≥ the use of Mo has no influence on wave velocity and coupling factor.

Fig. 11. Displacement profile along the depth of the multilayer AlN/Mo/Si, khAlN=2.7

SAW Parameters Analysis and Equivalent Circuit of SAW Device 453

Another equivalent model is based on the Coupling-Of-Modes (COM) theory. An excellent recent review of COM theory used in SAW devices was written by K.Hashimoto [10]. Based on the COM equations, as the force and voltage analogy can be used, the relationships between the terminal quantities at the one electrical port and two acoustic ports for an IDT have been done. K.Nakamura [44] introduced a simple equivalent circuit for IDT based on

In conclusion, the equivalent-circuit model is chosen because it can allow fast design. This allows the designer to determine the major dimensions and parameters in number of fingers, fingers width, aperture, delay line distance, frequency response, impedance

Based on Berlincourt et al [39] about equivalent circuit for Length Expander Bar with parallel electric field and with perpendicular electric field and based on the equivalent circuit for electromechanical transducer presented by Mason [36], Smith and al [41] have developed the equivalent circuit for IDT composed of N periodic sections of the form shown

One periodic section as shown in Figure 13 (a) can be presented by analogous onedimensional configurations: "crossed-field" model as in Figure 13 (b), and "in-line" model as in Figure 13 (c). In "crossed-field" model, the applied electric field is normal to the acoustic propagation vector; while in "in-line field" model, the electric field is parallel to the

The important advantage of two one-dimensional models is that each periodic section can be represented by equivalent circuit of Mason, as shown in Figure 14 for "crossed-field" model and Figure 15 for "in-line field" model. The difference between these two equivalent

<sup>0</sup> 4 2

ω

= = (12)

θ πω

circuits is that in "crossed-field" model, the negative capacitors are short-circuited.

α

COM approach that is developed in section 4.

Fig. 12. Interdigital transducer diagram

propagation vector.

Where:

in Figure 12.

parameters and transfer characteristics of SAW device.

**3.2 Equivalent circuit for IDT including N periodic sections** 
