**4. Modeling methods**

sents the relationship between input and output electrical signal. *H(f)* is a complex num‐ ber which can be defined as *H(f)* = *Aejφ*, being *A* = |*Vout/Vin|* the amplitude and *φ* the *phase shift* between *Vout* and *Vin*. In terms of voltage, the *insertion loss* (*IL*) in dB is given by 20 log10(*A*). Figure 3b, presents the frequency response of an AT-cut Z' propagating/SiO2 LW

**Figure 3.** a) Scheme of a LW delay line. It consists of two ports. In the input IDT an rf signal is applied which launches an acoustic propagating wave. The output signal is recorded at the output IDT. *D* is the distance between input and output IDT and *L* is the center-to-center distance between the IDTs. b) Frequency response of a LW device designed a fabricated by the authors of this chapter. The phase shift (dotted line) and IL (solid line) were measured using a net‐

In biosensors, biochemical interactions at the sensing area will modify the thickness and properties of the coating, and therefore variations in the amplitude and phase of the electri‐ cal transfer function can be measured. These variations can be monitored in real time, which

The LW delay line can be used as frequency determining element of an oscillator circuit (*closed loop* configuration). Effectively, in an oscillator circuit the LW device is placed as a de‐ lay line in the feedback loop of an rf amplifier in a closed loop configuration [10,45]. There‐ fore, a change in the wave velocity, due to a sensing effect, produces a time delay in the signal through the LW device which appears as phase-shift; this phase-shift is transferred in terms of frequency-shift in an oscillator configuration. The oscillator is, apparently, the sim‐ plest electronic setup: the low cost of their circuitry as well as the integration capability and continuous monitoring are some features which make the oscillators an attractive configura‐ tion for the monitoring of the determining parameter of the resonator sensor, which in the case of the LW device is the phase-shift of the signal at resonance [46-49]. However, due to the following drawbacks, in our opinion, the oscillators are not the best option for acoustic wave sensor characterization: 1) they do not provide direct information about signal ampli‐ tude; 2) they, eventually, can stop oscillation if insertion losses exceed the amplifier gain during an experiment; and 3) despite of the apparent simple configuration, a very good de‐ sign is necessary to guarantee that a LW resonator will operate at a specific frequency, and this is not a simple task. In effect, in the same way than in QCM oscillators it is required to assure that the sensor resonates on one defined resonance mode and does not "jump" be‐

device designed a fabricated by the authors of this chapter.

286 State of the Art in Biosensors - General Aspects

provides valuable information about the interaction process.

work analyzer.

As mentioned, there is an open research field regarding the employed materials, and its physical and geometric properties for achieving more optimized LW devices for biosensors applications. For instance, the thickness of the Love-wave guiding layer is a crucial parame‐ ter that can be varied to achieve a more sensitivity device. The fabrication of LW sensors is complex and expensive due to their micro sizes [54]; therefore, simulations and modeling of LW devices as previous steps to their production could be very valuable. Models allow re‐ lating changes in some characteristics of the wave, as the velocity, with changes in the physi‐ cal properties of the layers deposited over the sensing area, and thus, provide information about the sensing event. Nevertheless, modeling LW devices commonly requires simplified assumptions or the use of numerical methods [23] due to the complex nature of SAW propa‐ gation in anisotropic and piezoelectric materials.

In this section, information regarding the current most popular models used for modeling LW sensors is provided: the transmission line model, the dispersion equation and the Finite Element Method.

#### **4.1. Transmission line model**

It is well known that the propagation and attenuation of acoustic waves in guiding struc‐ tures can be obtained by equivalent transmission line models [8,55]. The theory of sound wave propagation is very similar mathematically to that of electromagnetic waves, so tech‐ 26 The solutions for these equations are given by:

22 differential equations are obtained:

1 **4.1. Transmission line model** 

8 guided waves [8].

*c c*

*Z Z*

10 Book Title

 It is well known that the propagation and attenuation of acoustic waves in guiding structures can be obtained by equivalent transmission line models [8,55]. The theory of sound wave propagation is very similar mathematically to that of electromagnetic waves, so techniques from transmission line theory are also used to build structures to conduct acoustic waves; and these are also called transmission lines. The *transmission line model* (TLM) for acoustic waves take advantage of the concepts and techniques of proven value in electromagnetic microwaves to corresponding problems in elastic

 A transmission line is characterized by its *secondary parameters* which are the propagation *wavevector k* (when scalar *wavenumber*) and the *characteristic impedance Zc* 10 (see Figure 4a). It is important to mention that these parameters do not depend on the transmission line length. In each plane of an electric transmission line it is possible to define a magnitude for a voltage and other for the current in the line. For acoustics transmission lines current and voltage in electromagnetic are replaced by the *particle velocity vp* and the *stress* -*TJ* 14 , respectively, where *J* indicates de stress direction (*J* = 1, 2, ..., 6) [55]. In an acoustic transmission line *Zc* 15 represents the relation between the stress *-TJ* and the particle velocity *vp* 16 of the material, and *k* quantifies how the wave energy will be propagated along the transmission line. To quantify the variations of *-TJ* and *vp* 17 when the wave propagates through the transmission line, the lumped elements models presented in Figure 4b and 4c are introduced. Figure 4b corresponds to the series model, and Figure 4c to the parallel model. The lumped elements of these models are called the transmission line *primary parameters*, which are dependent on the line length. Analyzing the parallel model of Figure 4c, the following coupled

*dr d v*

*dr*

( )

*J*

23 (0.1)

*d T <sup>Z</sup> Y v*

24 where *Z*=*jωL*, *Y*=*G*+*jωC* and *ω=*2*π/f.* Being *Z, L, Y, C, G, f*, the impedance, inductance, admittance,

*r r JJ J*

*T T*

*Tr Te Te*

*p*

capacitance and conductance per unit of length, respectively and *f* the frequency of *TJ* and *vp* 25 *.*

*p*

*J*

 

> 

*Z Y T*

Figure 4. a) Pictorial representation of a transmission line. b) Transmission line equivalent series model for acoustic propagation in a viscoelastic layer. c) Transmission line equivalent parallel model for acoustic propagation in a viscoelastic layer. **Figure 4.** a) Pictorial representation of a transmission line. b) Transmission line equivalent series model for acoustic propagation in a viscoelastic layer. c) Transmission line equivalent parallel model for acoustic propagation in a viscoe‐ lastic layer.

niques from transmission line theory are also used to build structures to conduct acoustic waves; and these are also called transmission lines. The *transmission line model* (TLM) for acoustic waves take advantage of the concepts and techniques of proven value in electro‐ magnetic microwaves to corresponding problems in elastic guided waves [8].

A transmission line is characterized by its *secondary parameters* which are the propagation *wavevector k* (when scalar *wavenumber*) and the *characteristic impedance Zc* (see Figure 4a). It is important to mention that these parameters do not depend on the transmission line length. In each plane of an electric transmission line it is possible to define a magnitude for a volt‐ age and other for the current in the line. For acoustics transmission lines current and voltage in electromagnetic are replaced by the *particle velocity vp* and the *stress* -*TJ* , respectively, where *J* indicates de stress direction (*J* = 1, 2,..., 6) [55]. In an acoustic transmission line *Zc* represents the relation between the stress *-TJ* and the particle velocity *vp* of the material, and *k* quantifies how the wave energy will be propagated along the transmission line. To quanti‐ fy the variations of *-TJ* and *vp* when the wave propagates through the transmission line, the lumped elements models presented in Figure 4b and 4c are introduced. Figure 4b corre‐ sponds to the series model, and Figure 4c to the parallel model. The lumped elements of these models are called the transmission line *primary parameters*, which are dependent on the line length. Analyzing the parallel model of Figure 4c, the following coupled differential equations are obtained:

$$\begin{aligned} \frac{d^2 T\_f}{dr^2} &= Z \cdot Y \cdot v\_p\\ \frac{d^2 v\_p}{dr^2} &= Z \cdot Y \cdot T\_f \end{aligned} \tag{1}$$

where *Z*=*jωL*, *Y*=*G*+*jωC* and *ω=*2*πf.* Being *Z, L, Y, C, G, f*, the impedance, inductance, admit‐ tance, capacitance and conductance per unit of length, respectively and *f* the frequency of *TJ* and *vp.*

The solutions for these equations are given by:

$$\begin{aligned} T\_f(\vec{r}) &= T\_f^+ e^{-\gamma \cdot \vec{r}} + T\_f^- e^{\gamma \cdot \vec{r}} \\ \upsilon\_p(\vec{r}) &= \frac{T\_f^+}{Z\_c} e^{-\gamma \cdot \vec{r}} - \frac{T\_f^-}{Z\_c} e^{\gamma \cdot \vec{r}} \end{aligned} \tag{2}$$

where *T+* and *T* are arbitrary values for the intensity of the incident an reflected waves, re‐ spectively. The *linear propagation exponent* or *complex propagation factor γ* is directly related to the wavevector (*γ*=*jk*) and is given by *γ =* (*ZY*) 1/2 *= α* + *jβ*. The real part of *γ,* denoted as *at‐ tenuation coefficient α*, represents the attenuation suffered by the wave when propagating through the transmission line. The imaginary part of *γ*, denoted as *phase coefficient β*, when multiplied by a distance, quantifies the phase shift that the wave suffers when traveling that distance. The characteristic impedance of the line *Zc* is given by *Zc*=(*Z*/*Y*) 1/2. First row of Ta‐ ble 3 shows the relationship between primary and secondary parameters of a transmission line for the series and parallel model.

The relation between the secondary (or primary) parameters and the properties of the trans‐ mission line materials is achieved by comparing Eqs. (1) with the motion equations of the LW assuming isotropic layers [55]. These relations are given in the second row of Table 3 for the series and parallel model.


**Table 3.** Equivalent transmission line model parameters in terms of the layer properties.

niques from transmission line theory are also used to build structures to conduct acoustic waves; and these are also called transmission lines. The *transmission line model* (TLM) for acoustic waves take advantage of the concepts and techniques of proven value in electro‐

*C* **dr**

C dr G dr

*p*

*J*

 

> 

*Z Y T*

*G* **dr**

*L/2* **dr** *L/2* **dr**

L/2 dr L/2 dr

b) b.2)

**r r + dr**

Figure 4. a) Pictorial representation of a transmission line. b) Transmission line equivalent series model for acoustic propagation in a viscoelastic layer. c) Transmission line equivalent parallel model for acoustic

**Figure 4.** a) Pictorial representation of a transmission line. b) Transmission line equivalent series model for acoustic propagation in a viscoelastic layer. c) Transmission line equivalent parallel model for acoustic propagation in a viscoe‐

r r + dr

*vp*

( )

( )

*vp* (r,t)

*dr d v*

*dr*

*J*

23 (0.1)

*d T <sup>Z</sup> Y v*

24 where *Z*=*jωL*, *Y*=*G*+*jωC* and *ω=*2*π/f.* Being *Z, L, Y, C, G, f*, the impedance, inductance, admittance,

*r r JJ J*

*Tr Te Te*

27 (0.2)

*J J r r <sup>p</sup> c c*

*T T vr e e Z Z*

*p*

capacitance and conductance per unit of length, respectively and *f* the frequency of *TJ* and *vp* 25 *.*

+

*TJ*

*TJ* (r,t)


10 Book Title

 It is well known that the propagation and attenuation of acoustic waves in guiding structures can be obtained by equivalent transmission line models [8,55]. The theory of sound wave propagation is very similar mathematically to that of electromagnetic waves, so techniques from transmission line theory are also used to build structures to conduct acoustic waves; and these are also called transmission lines. The *transmission line model* (TLM) for acoustic waves take advantage of the concepts and techniques of proven value in electromagnetic microwaves to corresponding problems in elastic

 A transmission line is characterized by its *secondary parameters* which are the propagation *wavevector k* (when scalar *wavenumber*) and the *characteristic impedance Zc* 10 (see Figure 4a). It is important to mention that these parameters do not depend on the transmission line length. In each plane of an electric transmission line it is possible to define a magnitude for a voltage and other for the current in the line. For acoustics transmission lines current and voltage in electromagnetic are replaced by the *particle velocity vp* and the *stress* -*TJ* 14 , respectively, where *J* indicates de stress direction (*J* = 1, 2, ..., 6) [55]. In an acoustic transmission line *Zc* 15 represents the relation between the stress *-TJ* and the particle velocity *vp* 16 of the material, and *k* quantifies how the wave energy will be propagated along the transmission line. To quantify the variations of *-TJ* and *vp* 17 when the wave propagates through the transmission line, the lumped elements models presented in Figure 4b and 4c are introduced. Figure 4b corresponds to the series model, and Figure 4c to the parallel model. The lumped elements of these models are called the transmission line *primary parameters*, which are dependent on the line length. Analyzing the parallel model of Figure 4c, the following coupled

A transmission line is characterized by its *secondary parameters* which are the propagation *wavevector k* (when scalar *wavenumber*) and the *characteristic impedance Zc* (see Figure 4a). It is important to mention that these parameters do not depend on the transmission line length. In each plane of an electric transmission line it is possible to define a magnitude for a volt‐ age and other for the current in the line. For acoustics transmission lines current and voltage

where *J* indicates de stress direction (*J* = 1, 2,..., 6) [55]. In an acoustic transmission line *Zc* represents the relation between the stress *-TJ* and the particle velocity *vp* of the material, and *k* quantifies how the wave energy will be propagated along the transmission line. To quanti‐

lumped elements models presented in Figure 4b and 4c are introduced. Figure 4b corre‐ sponds to the series model, and Figure 4c to the parallel model. The lumped elements of these models are called the transmission line *primary parameters*, which are dependent on the line length. Analyzing the parallel model of Figure 4c, the following coupled differential

*p*

*ZYv*

=××

*ZYT*

where *Z*=*jωL*, *Y*=*G*+*jωC* and *ω=*2*πf.* Being *Z, L, Y, C, G, f*, the impedance, inductance, admit‐ tance, capacitance and conductance per unit of length, respectively and *f* the frequency of *TJ*

=××

*J*

and *vp* when the wave propagates through the transmission line, the

, respectively,

*C* **dr**

**r r + dr**

r r + dr

*G* **dr**

C dr G dr

*L* **dr**

*vp*

*vp* (r,t) L dr

*TJ*


+

c)

*TJ* (r,t)

(1)

magnetic microwaves to corresponding problems in elastic guided waves [8].

in electromagnetic are replaced by the *particle velocity vp* and the *stress* -*TJ*

*d T*

*dr d v*

*dr*

The solutions for these equations are given by:

*J*

*p*

fy the variations of *-TJ*

a)

lastic layer.

*TJ k, Zc* (r,t)

*vp* (r,t)

22 differential equations are obtained:

1 **4.1. Transmission line model** 

8 guided waves [8].

r r + dr

26 The solutions for these equations are given by:

288 State of the Art in Biosensors - General Aspects

propagation in a viscoelastic layer.

equations are obtained:

and *vp.*

Figure 5a shows a simplify description in which a LW propagates in a waveguide structure. Two assumptions were made: 1) the materials in the figure are isotropic (for piezoelectric substrates this assumption is valid because of a low anisotropy) and 2) the main wave prop‐ agating in the *z* direction results from the interaction of two partial waves with the same component in *z* direction and opposites components in *y* direction [12].

If all the properties of the layers which are involved in the LW transmission line are known, it is possible to obtain the *phase velocity of the Love mode vφ*. The LW propagates in each layer *i* of the device in two directions *z* and *y*. In the case of a typical biosensor the de‐ vice consists of 5 layers with the subscripts *i* equal to: *S* for the substrate, *L* for the guiding layer, *SA* for the sensing area, *C* for the coating and *F* for the fluid. Direction *z* is known as the *longitudinal direction* and direction *y* as the *transverse direction*. The wave does not find

**Figure 5.** a) Simplified description of a LW traveling in a guided structure. b) Schematic representation of the wave‐ vectors *k1* and *k2* in two different media.

any material properties change in the longitudinal direction; hence it propagates in this di‐ rection acquiring a phase shift and attenuation (in case of material with losses). However, in the transverse direction a stationary wave exists when the resonance condition is met. Thus, each layer counts with two transmission lines, one in the transverse direction and the other in the longitudinal direction. When a wave propagates in *y'* direction (see Figure 5a), as it happens with the partial waves of a layer *i*, *ki* has that same direction (see Figure 5b), and therefore, it has components in *z* and *y* directions. In this way, the secondary parame‐ ters of each transmission line are determined from the projection of the parameter in the proper direction. The relation between the wavevector and the wave velocity is *ki* =*ω*/*vi* . Therefore, *ki* and *vi* have the same directions, so the wave velocity in *y'* direction also has components in *z* and *y*.

Equations (3) and (4) give the expression of the secondary parameters and wave velocity in the transverse and longitudinal directions, respectively:

$$Z\_{ciy} = Z\_{ci} \cos \theta\_{i'} \qquad k\_{iy} = \left| k\_i \right| \cos \theta\_{i'} \qquad v\_{iy} = \left| v\_i \right| \cos \theta\_i \tag{3}$$

$$\mathbf{Z}\_{c\dot{x}z} = \mathbf{Z}\_{c\dot{t}} \sin \theta\_{\mathbf{i}}, \qquad k\_{\dot{x}} = \begin{vmatrix} k\_i \end{vmatrix} \sin \theta\_{\mathbf{i}}, \qquad \upsilon\_{\dot{x}} = \left| \upsilon\_i \right| \sin \theta\_{\mathbf{i}} \tag{4}$$

The angles *θ<sup>i</sup>* are called *complex coupling angles* and depend on the material properties of each layer, but also on the material properties of the adjacent layers, since in each interface the Snell's laws have to be satisfied. In isotropic solids, the incident and reflected waves must all have the same component of *ki* in the longitudinal direction [55]; therefore:

$$k\_{\rm Sz} = k\_{\rm S} \sin \theta\_{\rm S} = k\_{L} \sin \theta\_{L} = k\_{\rm Lz} = k\_{\rm SA} \sin \theta\_{\rm SA} = k\_{\rm SAz} = k\_{\rm C} \sin \theta\_{\rm C} = k\_{\rm Cz} = k\_{\rm F} \sin \theta\_{\rm F} = k\_{\rm Fz} \tag{5}$$

These conditions, together with the *transverse resonance relation* [8] obtained from the trans‐ mission line models in the direction of resonance (transverse direction) (see Figure 6) pro‐ vides the phase velocity of the Love wave *vφ*.

The transverse resonance relation establishes that:

any material properties change in the longitudinal direction; hence it propagates in this di‐ rection acquiring a phase shift and attenuation (in case of material with losses). However, in the transverse direction a stationary wave exists when the resonance condition is met. Thus, each layer counts with two transmission lines, one in the transverse direction and the other in the longitudinal direction. When a wave propagates in *y'* direction (see Figure 5a), as it happens with the partial waves of a layer *i*, *ki* has that same direction (see Figure 5b), and therefore, it has components in *z* and *y* directions. In this way, the secondary parame‐ ters of each transmission line are determined from the projection of the parameter in the proper direction. The relation between the wavevector and the wave velocity is *ki*

**Figure 5.** a) Simplified description of a LW traveling in a guided structure. b) Schematic representation of the wave‐

*θL*

*<sup>y</sup> y'* Partial wave 1

Partial wave 2

a) b)

*z*

Therefore, *ki* and *vi* have the same directions, so the wave velocity in *y'* direction also has

Equations (3) and (4) give the expression of the secondary parameters and wave velocity in

 q *y*

*θ1 k1z*

*k1y k1* *z*

Medium 1

Medium 2

*k2y*

*k2*

*k2z*

*θ2*

 q

in the longitudinal direction [55]; therefore:

 q (3)

(4)

 q

(5)

cos , cos , cos *ZZ kk vv ciy ci i iy i i iy i i* == = qq

sin , sin , sin *ZZ kk vv ciz ci i iz i i iz i i* == = qq

sin sin sin sin sin *Sz S S L L Lz SA SA SAz C C Cz F F Fz kk k kk k k kk k* = = == = = == =

 q

each layer, but also on the material properties of the adjacent layers, since in each interface the Snell's laws have to be satisfied. In isotropic solids, the incident and reflected waves

are called *complex coupling angles* and depend on the material properties of

components in *z* and *y*.

Guiding layer

*θ<sup>L</sup> θ<sup>L</sup>*

Vacuum

290 State of the Art in Biosensors - General Aspects

Substrate

vectors *k1* and *k2* in two different media.

The angles *θ<sup>i</sup>*

must all have the same component of *ki*

qq

the transverse and longitudinal directions, respectively:

=*ω*/*vi* .

$$
\vec{Z}(P) + \vec{Z}(P) = 0 \tag{6}
$$

where *Z* <sup>→</sup> represents the acoustic impedance seen by the wave at the right of the plane *P* and *Z* <sup>←</sup> the acoustic impedance seen by the wave at the left of the plane. This relation states that the acoustic impedances looking both ways from some reference plane *P* must sum to zero. The location of *P* is arbitrary. The solution of applying this condition provides an angle *θ<sup>i</sup>* for one layer, depending on where *P* was located. From the material properties of each layer and applying the Snell's laws the angles for the rest of the layers can be found. Once these angles are known the *z* and *y* components of *ki* and *vi* can be obtained, and thus the phase velocity *vφ*. Assuming that almost all the energy of the wave is confined in the waveguide, the phase velocity can be defined as the wavefront velocity of the acoustic wave propagat‐ ing in the guiding layer, which in this case propagates in the *z* direction. In lossless materials *kiz* and *kiy* are real numbers and therefore *vφ* is given by:

$$\upsilon v\_{\phi} = \frac{\alpha}{k\_{Lz}} = \frac{\alpha}{\left| k\_{L} \right| \sin \theta\_{L}} \tag{7}$$

**Figure 6.** Equivalent transmission line model of the LW layered structure in the direction of resonance *y*. The lines are connected in series to satisfy the boundary conditions and the two semi-infinite layers are loaded with its characteris‐ tic impedance.

When the material has losses, *kiz* and *kiy* are complex numbers with real and imaginary parts, and then the attenuation coefficients appear:

$$k\_i = k\_{iz}\hat{z} + k\_{iy}\hat{y} = \left(\mathcal{J}\_{iz} - j\alpha\_{iz}\right)\hat{z} + \left(\mathcal{J}\_{iy} - j\alpha\_{iy}\right)\hat{y} \tag{8}$$

In this case, the phase velocity is given by:

$$\left| \upsilon\_{\phi} = \frac{\alpha \nu}{\mathcal{B}\_{Lz}} = \frac{\alpha}{\Re\left\{ k\_{Lz} \right\}} = \frac{\alpha \nu}{\Re\left\{ \left| k\_{L} \right| \sin \theta\_{L} \right\}} = \frac{1}{\Re\left\{ \frac{\sin \theta\_{L}}{\left| \upsilon\_{L} \right|} \right\}} \tag{9}$$

On the other hand, the *attenuation of the Love Wave αLW* is considered to happen mostly in the propagation direction *z*, since in the resonance direction, *y*, a stationary wave takes place. Therefore:

$$
\alpha\_{LW} = \alpha\_{Lz} = -\mathfrak{T}\{k\_{Lz}\} = -\mathfrak{T}\{\left|k\_L\right|\sin\theta\_L\}\tag{10}
$$

Thus, following this procedure it is possible to obtain the phase velocity and attenuation of a LW propagating in a layer. Nevertheless, to complete this, it is necessary to know the mate‐ rial properties of all the layers which integrate the LW device. However, when the device is used as a sensor, and in particular in biosensor applications, the coating layer properties are unknown parameters. Thus, quantifying the variations suffered by the mechanical and geo‐ metric layer properties over the sensing layer when measuring the electrical parameters is what is really interesting.

Variations in amplitude and phase of the transfer function *H(f)* = *Vout/Vin* (due to perturba‐ tions in the acoustic wave) can be monitored in real-time. These perturbations can occur due to variations of the mechanical and geometrical properties of the layers deposited over the sensing area. Such physical changes affect the propagation factor of the wave, and thus, the attenuation and phase velocity of the Love Wave. Next, the relations between LW electrical parameters defined in Section 3 (*φ* and *IL*) and the complex propagation factor are ex‐ plained.

The relation between the output and input voltage in a delay line (DL) of length *z* can be modeled by its transfer function *HDL(f)* in the following way:

$$H\_{DL}\left(f\right) = e^{-\gamma z} \tag{11}$$

where *γ* corresponds to the propagation factor of the wave in the line, which in this case cor‐ responds to that of the guiding layer in the *z* direction *γLz*. Assuming that the transfer func‐ tion of the input and output IDTs is equal to unit [26], the relation between the electrical signal in the output and input IDTs *H(f)* is the same than the one for the delay line:

$$H\left(f\right) = \frac{V\_{out}}{V\_{in}} = \frac{\left|V\_{out}\right|}{\left|V\_{in}\right|}e^{j\left(\phi\_{out} - \phi\_{in}\right)} = \frac{\left|V\_{out}\right|}{\left|V\_{in}\right|}e^{j\phi} = e^{-\gamma\_{1z}z} = e^{-(a\_{1z} + j\beta\_{1z})z}\tag{12}$$

Thus, taking into account the relations defined in Section 3, the normalized phase shift and *IL* are given by:

$$\begin{aligned} \frac{dL}{dz} &= -\alpha\_{Lz} 20 \log e\\ \frac{d\overline{\rho}}{dz} &= -\beta\_{Lz} \end{aligned} \tag{13}$$

The increment in *IL/z* and *φ/z* from a non perturbed state *γLz0* to a perturbed state *γLz1* is the following:

$$\begin{aligned} \Delta \frac{\text{IL}}{z} &= \left(\alpha\_{Lz1} - \alpha\_{Lz0}\right) \cdot 20 \log e \\ \Delta \frac{\text{op}}{z} &= \beta\_{Lz1} - \beta\_{Lz0} \end{aligned} \tag{14}$$

The last set of equations provides a relation between the experimental data and the physical parameters of the layers. The extraction of the layers physical parameters is a major prob‐ lem. Assuming that the physical properties of the substrate, guiding layer, gold and fluid medium are known and that these properties do not change during the sensing process, which can be the case in biosensing, still the parameters of the coating layer are not known. In comparison, the wave propagation direction in QCM coincides with the resonant direc‐ tion. Therefore, for low frequency QCM applications it is possible to assume that the bio‐ chemical interaction is translated to simple mass changes, since it is reasonable to assume that the thickness of the coating layer is acoustically thin. This simplifies enormously the pa‐ rameters extraction. However, in LW sensors, this assumption is not valid, and then the only two experimental data obtained in LW devices (Eq. (14)) are not enough to extract the un‐ known parameters of the coating. This, together with the complex equations which relate the measured data with the material properties, results in a complex issue that, to our knowledge, is not yet solved. Hence, there is an open research field of great interest related to this issue.

#### **4.2. Dispersion equation**

{ } { }

*k k*

*v*

292 State of the Art in Biosensors - General Aspects

Therefore:

plained.

what is really interesting.

j

b

a

 a

modeled by its transfer function *HDL(f)* in the following way:

ww

*Lz Lz L L* sin sin *<sup>L</sup>*

On the other hand, the *attenuation of the Love Wave αLW* is considered to happen mostly in the propagation direction *z*, since in the resonance direction, *y*, a stationary wave takes place.

{ } { } sin *LW Lz Lz L L*

Thus, following this procedure it is possible to obtain the phase velocity and attenuation of a LW propagating in a layer. Nevertheless, to complete this, it is necessary to know the mate‐ rial properties of all the layers which integrate the LW device. However, when the device is used as a sensor, and in particular in biosensor applications, the coating layer properties are unknown parameters. Thus, quantifying the variations suffered by the mechanical and geo‐ metric layer properties over the sensing layer when measuring the electrical parameters is

Variations in amplitude and phase of the transfer function *H(f)* = *Vout/Vin* (due to perturba‐ tions in the acoustic wave) can be monitored in real-time. These perturbations can occur due to variations of the mechanical and geometrical properties of the layers deposited over the sensing area. Such physical changes affect the propagation factor of the wave, and thus, the attenuation and phase velocity of the Love Wave. Next, the relations between LW electrical parameters defined in Section 3 (*φ* and *IL*) and the complex propagation factor are ex‐

The relation between the output and input voltage in a delay line (DL) of length *z* can be

where *γ* corresponds to the propagation factor of the wave in the line, which in this case cor‐ responds to that of the guiding layer in the *z* direction *γLz*. Assuming that the transfer func‐ tion of the input and output IDTs is equal to unit [26], the relation between the electrical

( ) *<sup>z</sup> H fe DL* g

signal in the output and input IDTs *H(f)* is the same than the one for the delay line:

*in in in*

*V V V H f e ee e V V V* j j-

( ) ( ) ( ) *out in Lz Lz Lz out <sup>j</sup> out <sup>j</sup> <sup>z</sup> j z out*

j g

== = = = (12)

 - + a b

= = -Á = -Á *k k*

== = = <sup>Â</sup> <sup>Â</sup> ì ü ï ï Âí ý

 w

> q

1

*L*

ï ï î þ

q

(9)

*v*

q

(10)

= (11)

The *dispersion equation* provides the wave phase velocity as a function of the guiding lay‐ er thickness. The procedure for obtaining this equation for a two-layer system (guiding layer and substrate) is detailed in reference [56]. Broadly, this equation is reached after imposing the boundary conditions to determine the constants appearing in the particle displacement expressions of the waveguide and the substrate. These displacements are the solution of the equation of motion in an isotropic and non-piezoelectric material*.* Af‐ ter extensive algebraic manipulation [56], the dispersion equation for a two-layer system is found, resulting [57]:

$$\tan\left(k\_{Ly}d\right) = \frac{\mu\_S}{\mu\_L} \sqrt{\frac{1 - \left(v\_{\phi}^2 \Big/ v\_S^2\right)}{\left(v\_{\phi}^2 \Big/ v\_L^2\right) - 1}}\tag{15}$$

where *kLy* is the guiding layer transverse wavenumber in *y* direction, given by:

$$k\_{Ly} = \sqrt{\frac{\alpha^2}{v\_L^2} - \frac{\alpha^2}{v\_\varphi^2}}\tag{16}$$

Taking into account the relation between the frequency and wave wavelength, *f = vφ/λ*, the argument of the tangent in Eq.(15) can be written as:

$$k\_{Ly}d = 2\pi v\_{\varphi} \frac{d}{\lambda} \sqrt{\frac{1}{v\_L^2} - \frac{1}{v\_{\varphi}^2}}\tag{17}$$

where the ratio *d/λ* is the normalized guiding layer thickness.

From the dispersion equation, the phase velocity can be solved numerically. On the other hand, the *group velocity*, *vg,* as a function of the normalized guiding layer thickness can also be determined from the phase velocity by the formula [34]:

$$\mathbf{v}\_{\rm g} = \mathbf{v}\_{\rm \phi} \left( \mathbf{l} + \frac{d/\lambda}{\mathbf{v}\_{\rm \phi}} \frac{\mathbf{d} \mathbf{v}\_{\rm \phi}}{\mathbf{d} \left( d/\lambda \right)} \right) \tag{18}$$

The phase velocity and group velocity of an AT-cut quartz/SiO2 Z' propagating layered structure were calculated using the dispersion equation (Eq. (15)) solved numerically through the bisection method. The data used to solve the equation for this LW structure were: *vS* = 5099 m/s, *ρS* = 2650 kg/m3 , *vL =* 2850 m/s, *ρ<sup>L</sup> =* 2200 kg/m3 and λ = 40 µm. The respective shear modulus were obtained through *µi* = *vi* <sup>2</sup> *ρ<sup>i</sup>* . Results are depicted in Fig‐ ure 7. This figure provides information about the changes in the wave phase velocity due to changes in the guiding layer thickness. Assuming that the perturbing mass layer de‐ posited over the sensing area is of the same material than the guiding layer, this equa‐ tion will provide information over the sensing event. Nevertheless, this assumption is far

from the biosensors reality, where the consideration of a five–layer model is required (see Figure 1b). 20 far from the biosensors reality, where the consideration of a five–layer model is required (see Figure

Running Title <sup>15</sup>

*v v*

*<sup>S</sup> <sup>S</sup>*

*L L*

2 2

2 2

d

*d v*

 

*v d*

The phase velocity and group velocity of an AT-cut quartz/SiO2 12 Z' propagating layered structure 13 were calculated using the dispersion equation (Eq. (0.15)) solved numerically through the bisection method. The data used to solve the equation for this LW structure were: *vS* 14 = 5099 m/s,

, *vL =* 2850 m/s, *ρL =* 2200 kg/m3 15 and λ = 40 μm. The respective shear modulus were

<sup>2</sup>*ρi* 16 . Results are depicted in Figure 7. This figure provides information about

19 layer, this equation will provide information over the sensing event. Nevertheless, this assumption is

*v v*

*L*

*v v*

4 Taking into account the relation between the frequency and wave wavelength, *f = vφ/λ*, the argument

1 1 <sup>2</sup> *Ly*

8 From the dispersion equation, the phase velocity can be solved numerically. On the other hand, the *group velocity*, *vg* 9 *,* as a function of the normalized guiding layer thickness can also be determined

*<sup>d</sup> kd v*

1 d *<sup>g</sup>*

11 (0.18)

*v v*

6 (0.17)

2 2

2 2

*v v*

1

1 (0.15)

2 2 *Ly L*

3 (0.16)

tan

5 of the tangent in Eq.(0.15) can be written as:

10 from the phase velocity by the formula [34]:

*ρS* = 2650 kg/m3

21 1b).

obtained through *μi* = *vi*

7 where the ratio *d/λ* is the normalized guiding layer thickness.

*Ly*

*k*

where *kLy* 2 is the guiding layer transverse wavenumber in *y* direction, given by:

*k d*

1

Figure 7. a) AT-cut Z' propagating quartz/SiO2 phase velocity (continuous line) and group velocity (dashed line) for λ = 40 μm. **Figure 7.** a) AT-cut Z' propagating quartz/SiO2 phase velocity (continuous line) and group velocity (dashed line) for λ = 40 μm.

Mc Hale *et al*. developed the dispersion equation of three and four-layer systems neglecting piezoelectricity of the substrate [33]. When the substrate piezoelectricity is not considered, the dispersion equation is simplified. This can be the case of quartz substrates, which piezo‐ electricity is low. However, as the piezoelectricity of a substrate increases (like in the case of LT and LN), neglecting piezoelectricity, or assuming it is accounted for a stiffening effect in the phase velocity, may be less valid [58]. Liu et al. provided a theoretical model for analyz‐ ing the LW in a multilayered structure over a piezoelectric substrate [58]. Nevertheless, in our opinion, for those applications with a high number of layers, as in the case of biosensors, the use of the TLM is more convenient, since it is a very structured and intuitive model where the addition of an extra layer does not make the procedure more complex. From the programming point of view, this is an enormous advantage.

The phase velocity provided by the dispersion equation can be used to determine the opti‐ mal guiding layer thickness, which provides a maximum sensitivity. This issue will be ad‐ dressed in Section 5.

#### **4.3. LW sensor 3D FEM simulations**

ter extensive algebraic manipulation [56], the dispersion equation for a two-layer system

( ) ( )

m

=

where *kLy* is the guiding layer transverse wavenumber in *y* direction, given by:

m

tan

argument of the tangent in Eq.(15) can be written as:

where the ratio *d/λ* is the normalized guiding layer thickness.

be determined from the phase velocity by the formula [34]:

respective shear modulus were obtained through *µi* = *vi*

were: *vS* = 5099 m/s, *ρS* = 2650 kg/m3

*Ly*

*k*

*k d*

1

( )

*v v* j

*L L*

2 2

*v v*j

Taking into account the relation between the frequency and wave wavelength, *f = vφ/λ*, the

2 2

( ) d

l

d j

The phase velocity and group velocity of an AT-cut quartz/SiO2 Z' propagating layered structure were calculated using the dispersion equation (Eq. (15)) solved numerically through the bisection method. The data used to solve the equation for this LW structure

ure 7. This figure provides information about the changes in the wave phase velocity due to changes in the guiding layer thickness. Assuming that the perturbing mass layer de‐ posited over the sensing area is of the same material than the guiding layer, this equa‐ tion will provide information over the sensing event. Nevertheless, this assumption is far

j

l

æ ö = + ç ÷ è ø *<sup>g</sup>*

j

*v v*

*L*

From the dispersion equation, the phase velocity can be solved numerically. On the other hand, the *group velocity*, *vg,* as a function of the normalized guiding layer thickness can also

*Ly* 2 2 *L*

1 1 <sup>2</sup> *Ly*

l

j

*<sup>d</sup> kd v*

p

1

j

w w


j

*S S*

2 2

2 2

*v v*

1


= - (16)

<sup>=</sup> - (17)

*<sup>d</sup> <sup>v</sup> v v v d* (18)

, *vL =* 2850 m/s, *ρ<sup>L</sup> =* 2200 kg/m3 and λ = 40 µm. The

. Results are depicted in Fig‐

<sup>2</sup> *ρ<sup>i</sup>*

is found, resulting [57]:

294 State of the Art in Biosensors - General Aspects

The models mentioned before, applied simplifying assumptions like considering the device substrate as isotropic and neglecting the substrate piezoelectricity. This makes the models far from the LW device reality. For a more accurate calculation of piezoelectric devices oper‐ ating in the sonic and ultrasonic range, numerical methods such as finite element method (FEM) or boundary element methods (BEM) are the preferred choices [59]. The combination of the FEM and BEM (FEM/BEM) has been used by many authors for the simulation of SAW devices [60]. Periodic Green's functions [61] are the basis of this model. However, the FEM/BEM only applies to infinite periodic IDTs and it is a 2 dimensional (2D) analysis. Thus, this method is only an approximation of a real finite SAW device.

Simulations of piezoelectric media require the complete set of fundamental equations relat‐ ing mechanical and electrical phenomena in 3 dimensions (3D). The 3D-FEM can handle these types of differential equations. The FEM formulation for piezoelectric SAW devices is well explained in [59]. In general, the procedure for simulating LW devices using the 3D-FEM is the following: 1) the 3D model of the device is build using a computer-aided design (CAD) software; 2) the 3D model is imported to a commercial finite element software, which allows piezoelectric analysis; 3) the material properties of the involved materials are introduced in the software; 4) the convenient piezoelectric finite element is selected; 5) the model is meshed with the selected finite element; and 6) the simulation is run in the soft‐ ware. As result the software obtains the particle displacements and voltage at every node of the model.

Although 3D-FEM simulations are extremely useful tools for studying LW device electroacoustic interactions, LW simulations in real size are still a challenge. Delay lines in practice are of many wavelengths and simulate them would require having too many finite ele‐ ments. Thus, further efforts are required in order to achieve simulations able to reproduce real cases, which do not consume excessive computational resources. Nevertheless, some authors have simulated scaled LW sensors using this method [62-65].

## **5. Sensitivity and limit of detection**

A key parameter when designing a LW biosensor is the device sensitivity [13]. In general terms, the sensitivity is defined as the derivative of the response (*R*) with respect to the physical quantity to be measured (*M*):

$$S = \lim\_{\Delta M \to 0} \frac{\Delta R}{\Delta M} = \frac{dR}{dM} \tag{19}$$

It is possible to have different units of sensitivity depending on the used sensor response. E.g. for frequency output sensors, frequency (*R = f*), relative frequency (*R = f / f0*), frequency shift (*R = f - f0*) and relative frequency shift (*R =* ( *f - f0* )/ *f0)* can be found, being *f0* the non perturbed starting frequency. Hence, four different possibilities of sensitivity formats are possible, and therefore it is extremely important to mention which case is been used in each application.

The sensitivity of LW sensors gives the correlation between measured electric signals deliv‐ ered by the sensor and a perturbing event which takes place on the sensing area of the sen‐ sor. A high sensitivity relates a strong signal variation with a small perturbation [32]. Depending on the used electronic configuration, the electrical signal measured in LW devi‐ ces can be: operation frequency, amplitude (or Insertion Loss), and phase. From this signal, and using the theoretical models, the phase velocity and group velocity can be obtained (see Sections 4.1 and 4.2).

devices [60]. Periodic Green's functions [61] are the basis of this model. However, the FEM/BEM only applies to infinite periodic IDTs and it is a 2 dimensional (2D) analysis.

Simulations of piezoelectric media require the complete set of fundamental equations relat‐ ing mechanical and electrical phenomena in 3 dimensions (3D). The 3D-FEM can handle these types of differential equations. The FEM formulation for piezoelectric SAW devices is well explained in [59]. In general, the procedure for simulating LW devices using the 3D-FEM is the following: 1) the 3D model of the device is build using a computer-aided design (CAD) software; 2) the 3D model is imported to a commercial finite element software, which allows piezoelectric analysis; 3) the material properties of the involved materials are introduced in the software; 4) the convenient piezoelectric finite element is selected; 5) the model is meshed with the selected finite element; and 6) the simulation is run in the soft‐ ware. As result the software obtains the particle displacements and voltage at every node

Although 3D-FEM simulations are extremely useful tools for studying LW device electroacoustic interactions, LW simulations in real size are still a challenge. Delay lines in practice are of many wavelengths and simulate them would require having too many finite ele‐ ments. Thus, further efforts are required in order to achieve simulations able to reproduce real cases, which do not consume excessive computational resources. Nevertheless, some

A key parameter when designing a LW biosensor is the device sensitivity [13]. In general terms, the sensitivity is defined as the derivative of the response (*R*) with respect to the

It is possible to have different units of sensitivity depending on the used sensor response. E.g. for frequency output sensors, frequency (*R = f*), relative frequency (*R = f / f0*), frequency shift (*R = f - f0*) and relative frequency shift (*R =* ( *f - f0* )/ *f0)* can be found, being *f0* the non perturbed starting frequency. Hence, four different possibilities of sensitivity formats are possible, and therefore it is extremely important to mention which case is been used in each

The sensitivity of LW sensors gives the correlation between measured electric signals deliv‐ ered by the sensor and a perturbing event which takes place on the sensing area of the sen‐ sor. A high sensitivity relates a strong signal variation with a small perturbation [32]. Depending on the used electronic configuration, the electrical signal measured in LW devi‐

<sup>D</sup> = = <sup>D</sup> (19)

0 lim *M R dR <sup>S</sup>* D ® *M dM*

Thus, this method is only an approximation of a real finite SAW device.

authors have simulated scaled LW sensors using this method [62-65].

**5. Sensitivity and limit of detection**

physical quantity to be measured (*M*):

of the model.

296 State of the Art in Biosensors - General Aspects

application.

When the sensor response *R* is the phase velocity, and the perturbing event which takes place on the sensing area is a variation of the *surface mass density* of the coating *σ* (*σ=h ρC*), the *velocity mass sensitivity Sv* <sup>σ</sup> of a LW device at a constant frequency is given by [32,66]:

$$S^{\upsilon}\_{\sigma} = \frac{1}{v\_{\varphi0}} \frac{\partial v\_{\varphi}}{\partial \sigma} \bigg|\_{f} \tag{20}$$

where *vφ0* is the unperturbed phase velocity and *v<sup>φ</sup>* is the phase velocity after a surface mass change.

Hence, *Sv* σ has units of m2 /kg. In Eq. (20) partial derivatives have been considered as the phase velocity depends on several variables. The surface mass sensitivity reported for LW devices in literature are between 150-500 cm2 /g [45,67].

In sensor applications the phase velocity shift must be obtained from the experimental val‐ ues of phase or frequency shifts. For the closed loop configuration, where the experimental‐ ly measured quantity is the frequency, the *frequency mass sensitivity Sf* <sup>σ</sup> is defined as:

$$S^f\_\sigma = \frac{1}{f\_0} \frac{df}{d\sigma} \tag{21}$$

Jakoby and Vellekoop [66] noticed that the sensitivity measured by frequency changes in an oscillator (Eq. (21)) differs from the estimated velocity mass sensitivity (Eq. (20)) by a factor *vg*/*vφ*, since Love modes are dispersive. Thus, a typical 10% difference can be noted between *Sf σ* and *Sv σ.*

For the open loop configuration, where the experimentally measured quantity is the phase, the *phase sensitivity* (also called *gravimetric sensitivity*) *S<sup>φ</sup>* σ in absence of interference is de‐ fined as:

$$S^{\rho}\_{\sigma} = \frac{1}{k\_{\perp z}D} \frac{d\rho}{d\sigma} = S^{v}\_{\sigma} \tag{22}$$

where *D* is the distance between input and output IDT (see Figure 3) and *kLz* is the wave‐ number of the Love mode, therefore *kLzD* is the unperturbed phase *φ0*.

The theoretical mass sensitivity of LW devices, derived from the perturbation theory[55], can be determined from the phase velocity according to [57]:

$$S\_{\sigma}^{\upsilon} = \frac{-1}{\rho\_L d} \left( 1 + \frac{\sin(k\_{Ly}d)\cos(k\_{Ly}d)}{k\_{Ly}d} + \frac{\rho\_S}{\rho\_L} \frac{\cos^2(k\_{Ly}d)}{k\_{Sy}d} \right)^{-1} \tag{23}$$

where *kLy* is the wavenumber in the guiding layer (Eq. (16)) and *kSy* is the wavenumber in the substrate (Eq. (24)). The minus sing in Eq. (23) indicates that the phase velocity of the pertur‐ bed event due to an increment in the surface mass density is less than the unperturbed phase velocity (Δ*v<sup>φ</sup> = vφ- vφ0*).

$$k\_{Sy} = \sqrt{\frac{\alpha^2}{\upsilon\_{\varphi}^2} - \frac{\alpha^2}{\upsilon\_S^2}}\tag{24}$$

The phase velocity can be obtained with the dispersion equation (see Figure 7) or transmis‐ sion line model. Once the phase velocity is known, the mass sensitivity curve can be ob‐ tained using Eq. (23). In Figure 8, the dependence of the sensitivity with the guiding layer thickness for an AT-cut quartz/SiO2 Z' propagating structure is depicted. A maximum mass sensitivity *Sv <sup>σ</sup>* of -39.44 m2 /kg is observed at *d*/λ of 0.171 which corresponds to a *d* = 6.84 µm. The phase velocity at *d*/λ = 0.171 is *v<sup>φ</sup>* = 4140 m/s (see Figure 7) which leads to a synchronous frequency *fs* = *vφ*/λ = 103.5 MHz.

**Figure 8.** a) AT-cut quartz Z' propagating /SiO2 mass sensitivity considering λ = 40 μm.

Thus, as mentioned previously, for very small thicknesses compared to the wavelength, the acoustic field is not confined to the surface and deeply penetrates into the piezoelectric sub‐ strate, resulting in low mass sensitivity [68]. If the thickness is increased the sensitivity rises, as the acoustic energy is more efficiently trapped in the guiding layer. However, an exces‐ sive increase in the guiding layer thickness leads to a reduction of wave energy density and the mass sensitivity decreases [69]. Therefore, there is an optimal guiding layer thickness at which a maximum mass sensitivity is achieved for a specific wavelength.

In those applications where a coating layer in contact with a liquid is deposited over the sensing area, such is the case for biosensors, both changes in the surface mass density *Δσ* and in the mass viscosity Δ(*ρCηC*) 1/2 of the coating occur, due to the biochemical interaction between the coating and the liquid medium. In this case, the sensitivity can be modeled by the four components matrix shown in Eq.(25) [15]. These components relate shifts in surface mass density and in mass viscosity to the measured electrical signals: *phase shift φ* and *Inser‐ tion Loss (IL)*. The matrix components relate shifts of surface density Δ*σ* and mass viscosity Δ(*ρη*)1/2 to shifts of electrical phase Δ*φ* and signal attenuation Δ*IL*. Notice that *Sφ,σ* is not the same than *S<sup>φ</sup>* σ in Eq.(22).

$$
\begin{bmatrix}
\Delta\boldsymbol{\rho} \\
\Delta\boldsymbol{L}
\end{bmatrix} = \begin{bmatrix}
\mathbb{S}\_{\boldsymbol{\rho},\boldsymbol{\sigma}} & \mathbb{S}\_{\boldsymbol{\rho},\sqrt{\boldsymbol{\rho}\_{\mathbb{C}}\boldsymbol{\eta}\_{\mathbb{C}}}} \\
\mathbb{S}\_{\boldsymbol{\Pi},\boldsymbol{\sigma}} & \mathbb{S}\_{\boldsymbol{\Pi},\sqrt{\boldsymbol{\rho}\_{\mathbb{C}}\boldsymbol{\eta}\_{\mathbb{C}}}} \\
\end{bmatrix} \begin{bmatrix}
\Delta\boldsymbol{\sigma} \\
\Delta\sqrt{\boldsymbol{\rho}\_{\mathbb{C}}\boldsymbol{\eta}\_{\mathbb{C}}} \\
\end{bmatrix} \tag{25}
$$

Generally, the high sensitivity of microacoustic sensors is closely related to the fact that they show a high temperature stability (low TCF) and a large signal-to-noise ratio, which, in turn yields low detection limits and a high resolution of the sensor [13]. The *limit of detection* (LOD) is a very important characteristic of acoustic biosensors, since it gives the minimum surface mass that can be detected by the device. It can be directly derived from the ratio be‐ tween the noise in the measured electrical signal *Nf* and the sensitivity of the device. For in‐ stance, in a closed loop configuration, this noise *Nf* is the RMS value of the frequency measured over a given period of time in stable and constant conditions [32]. It is usually rec‐ ommended to measure a signal variation higher than 3 times the noise level in order to con‐ clude from an effective variation [70]. From this recommendation, it comes out that the LOD is given by [32,71]:

$$\text{LOD} = \sigma\_{\text{min}} = \frac{3 \cdot N\_f}{S\_\sigma^f \cdot f} \tag{26}$$

where *f* is the operation frequency.

<sup>1</sup> <sup>2</sup> <sup>1</sup> sin( )cos( ) cos ( ) <sup>1</sup> *<sup>v</sup> Ly Ly <sup>S</sup> Ly L Ly L Sy*


*d kd k d*

è ø

where *kLy* is the wavenumber in the guiding layer (Eq. (16)) and *kSy* is the wavenumber in the substrate (Eq. (24)). The minus sing in Eq. (23) indicates that the phase velocity of the pertur‐ bed event due to an increment in the surface mass density is less than the unperturbed

2 2

 w

*S*

= - (24)

/kg is observed at *d*/λ of 0.171 which corresponds to a *d* = 6.84 µm.

*Sy* 2 2

w

*v v* j

The phase velocity can be obtained with the dispersion equation (see Figure 7) or transmis‐ sion line model. Once the phase velocity is known, the mass sensitivity curve can be ob‐ tained using Eq. (23). In Figure 8, the dependence of the sensitivity with the guiding layer thickness for an AT-cut quartz/SiO2 Z' propagating structure is depicted. A maximum mass

The phase velocity at *d*/λ = 0.171 is *v<sup>φ</sup>* = 4140 m/s (see Figure 7) which leads to a synchronous

Thus, as mentioned previously, for very small thicknesses compared to the wavelength, the acoustic field is not confined to the surface and deeply penetrates into the piezoelectric sub‐ strate, resulting in low mass sensitivity [68]. If the thickness is increased the sensitivity rises,

*k*

**Figure 8.** a) AT-cut quartz Z' propagating /SiO2 mass sensitivity considering λ = 40 μm.

*S*

298 State of the Art in Biosensors - General Aspects

phase velocity (Δ*v<sup>φ</sup> = vφ- vφ0*).

sensitivity *Sv*

s

*<sup>σ</sup>* of -39.44 m2

frequency *fs* = *vφ*/λ = 103.5 MHz.

r

*kd kd kd*

r

(23)

 r

> The LOD is improved by minimizing the influence of temperature on the sensor re‐ sponse [13]. The stability with respect to temperature can be achieved by implementing temperature control in the biosensor system and choosing materials of low TCF, as seen in Section 2.1.

#### **6. LW packaging and flow cells**

Flow cells are usually constructed with an inlet at one side, an outlet at the other, and the channels facing the sensor (see Figure 9). In most cases, a rubber seal is used for sealing, and in LW flow-cells additional absorbers made with rubber materials at the ends of the IDTs are recommended to improve the signal response (see Figure 9b). Factors such as the flow cell design, flow patterns and flow-through system influence the binding efficiency and the course of binding kinetics, resulting in possible variations of the true results [72].

The permittivity and the dielectric losses of the liquid medium, necessary in LW biosensors applications, influence the propagation of Love modes, since this medium acts as an addi‐ tional layer. When the liquid medium is deposited over the device surface, the presence of this medium over the IDTs modifies the electrodes transfer function [73]. Permittivity and dielectric losses of the liquid lead to a corresponding change of the IDT input admittance, which influences the amplitude of the signal. A flow–through cell is crucial to eliminate this electric influence of the liquid. Such flow-cell isolates the IDTs from the liquid, confining the liquid in the region between the IDTs (sensing area). LW device packaging or flow-cells gen‐ erally use walls to accomplish this purpose. These walls, when pressed onto the device sur‐ face, disturb the acoustic wave, resulting in an increase in overall loss and distortion of the sensor response. Walls must be designed to minimize the contact area in the acoustic path in order to obtain the minimum acoustic attenuation (see Figure 9c). It is known that the acous‐ tic wave is significantly affected when increasing the walls width [27] and that materials used for these walls play an important role. Hence, great care must be taken to ensure that the designed LW flow cells do not greatly perturb the acoustic signal. Recently, some re‐ searchers have explored different possibilities to achieve the packaging of LW sensors for fluidic applications [27,74] and other authors have being exploring different LW flow-cell approaches [53,75,76].

**Figure 9.** A developed LW flow cell for immunosensors application designed and fabricated by the authors of this chapter [53]. a) Overview of the flow cell b) Microscope view through the PMMA of the LW sensor and flow cell ele‐ ments. c) PDMS square seal with pick end to minimize the contact area.

Further improvements for LW biosensors flow cells can be achieved, since it is an entirely new field and the development trends moves towards smaller flow cells which allows the use of less analyte. For instance, investigation on the materials used for creating flow cells, packaging, and flow patterns in microfluidic channels [72] would enhance LW performance and move them faster to the lab-on-a-chip arena.
