**2. Piezoelectric filters**

This chapter is organised in three parts. First, a brief summary of the properties of various piezoelectric materials is presented. The second part describes the principal use of piezoelectric materials for RF applications: filters and resonators. The third part is dedicated to the new and emerging RF functions available owing to MEMS technology on piezoelectric materials:

For RF applications, two types of piezoelectric materials can be identified: those that will not be crossed by the RF signal and those that are in direct contact with the RF signal. For applications where the piezoelectric plate must operate in the gigahertz range, few materials can be used. The principal parameter for selecting a piezoelectric material is the means of production and compatibility with technological processes similar to those used in micro‐ electronics. Another parameter is the maximum frequency of operation with a good quality

A material is defined by several physical parameters, such as piezoelectric constants, stiffness, complex dielectric constants and other constants. End users of piezoelectric materials are more interested in other parameters, such as the electromechanical factor, the deposition process, which influences piezoelectric behaviour, the ability to hold up high RF power, and conse‐ quently nonlinear behaviour, the maximum frequency of operation. This list is not exhaustive and can be extended in accordance with use. Limited data about the characteristics of piezo‐ electric materials submitted to high RF power are available in the literature. Moreover, many results suggest that thin film exhibits better characteristics than bulk materials and that the coupling factor, the piezoelectric constant and other parameters are dependent on the fabri‐ cation process. It is difficult to obtain the absolute values of parameters, such as piezoelectric constants, mechanical stiffness and so on. Despite this, it is interesting to determine their important characteristics based on the available literature data to compare piezoelectric

values in the gigahertz range. These two parameters are often used as figures of merit. These materials are lead free except for PZT. This last material will probably be removed from the composition of devices according to regulations in several countries. Another important characteristic will be the tunability of the resonance of the piezoelectric plate. This will be a challenge for the future due to the great number of standards and the limited space in a mobile object. The ideal filter can be electrically tuned and adapted to several frequency bands. Some materials, such as barium strontium titanate (BST), have been used to realise a tunable filter. Although at this time, tunability is limited to a small percentage of the central frequency and can only compensate the variability of process fabrication. At this time, AlN offers the best trade‐off, but it cannot be electrically tuned. However, it is possible to improve the properties of AlN by substituting a part of Al with Sc; this mainly leads to an increase in piezoelectric

2

and the mechanical quality factor

switches, phase shifters and varactors.

factor.

206 Piezoelectric Materials

materials.

coefficients.

**1.2. Piezoelectric materials and key parameters**

**Table 2** shows the square values of the coupling factor *kt*

The first filter type was the SAW filter. Probably one of its most important characteristics is the accuracy of the resonant frequency of the structure. Microelectronic technology offers a good resolution to realise the structure in the plane of the substrate, but thickness control is more difficult. As thickness is crucial to bulk resonance mode, without a minimum of accuracy to control the thickness, the resonant frequency will sweep as it is directly linked to the thickness. This is probably one of the most important reasons to use a SAW filter as this type of filter does not require high accuracy for piezoelectric plate thickness. SAW filters require only the control of the dimensions of interdigital transducers (IDTs), particularly the finger's width and the shape of the finger that must exhibit very low deviation compared with the mean value. Nevertheless, SAW filters occupy a more important area than other filter types.

#### **2.1. Surface acoustic wave (SAW) filter**

The technological steps to realise a SAW transducer are restricted to the depositing of a metal layer that is then etched to obtain the final electrodes on the surface of the piezoelectric plate. This technology was first used for delay lines and for bandpass filters.

#### *2.1.1. Principle*

The basis of this type of filter is the generation of a well‐known SAW. Among SAWs, the Rayleigh wave that exhibits the same behaviour is preferable because for propagation direction and the wave velocity is independent of the frequency. Nevertheless, it implies that the propagation path has a thickness two times higher than the wavelength and this condition cannot be satisfied in any case. Thus, Lamb waves [14] are more often generated. This type of filter is based on a piezoelectric plate deposited on substrate (silicon, alumina and so on). A pair of IDT is etched at the top. One will be excited by the input signal, and the other one will receive the acoustic wave generated by the first one. **Figure 6** shows the basic requirements for a bandpass filter. The distance between two adjacent fingers must be half the wavelength (**Figure 7**). The frequency of operation and consequently the filter characteristics are dependent on the dimensions of IDTs. The higher the number of fingers of the transducer, the greater the selectivity of the filter and the more the bandwidth is reduced.

**Figure 6.** Basic SAW filter.

**Figure 7.** IDT specifications.

#### *2.1.2. Topologies*

On the basis of an IDT, several research studies were conducted to improve the characteristics of filters. It is possible to use different configurations by changing the way IDTs are coupled, by adding attenuators or reflectors at both ends of the surface of the piezoelectric plate. It is also possible to select specific materials to compensate the temperature variation of the piezoelectric layer and to minimise the characteristics of the shift [15].

The longitudinally coupled IDT shown in **Figure 8(a)** is probably the most used, but it requires reflectors to improve coupling or in other cases attenuators to reduce spurious wave propa‐ gation on the substrate surface [16]. It is also possible to use transverse coupling [17], but this configuration is more difficult to operate (**Figure 8(b)**). Finally, the association of several transmission SAW structures can be done as shown in **Figure 8(c)** to improve filter character‐ istics. However, the main drawback is the occupied surface that is increased by the number of SAW devices. Although SAW devices exhibit advantages, in terms of fabrication, their surface is a handicap. Nevertheless, this type of filter is always used in commercial products.

**Figure 8.** Main structures based on SAW: (a) longitudinal coupling; (b) transverse coupling; (c) ladder structure.

#### **2.2. Bulk acoustic wave (BAW) filters**

#### *2.2.1. Bulk acoustic wave resonators*

cannot be satisfied in any case. Thus, Lamb waves [14] are more often generated. This type of filter is based on a piezoelectric plate deposited on substrate (silicon, alumina and so on). A pair of IDT is etched at the top. One will be excited by the input signal, and the other one will receive the acoustic wave generated by the first one. **Figure 6** shows the basic requirements for a bandpass filter. The distance between two adjacent fingers must be half the wavelength (**Figure 7**). The frequency of operation and consequently the filter characteristics are dependent on the dimensions of IDTs. The higher the number of fingers of the transducer, the greater the

On the basis of an IDT, several research studies were conducted to improve the characteristics of filters. It is possible to use different configurations by changing the way IDTs are coupled, by adding attenuators or reflectors at both ends of the surface of the piezoelectric plate. It is also possible to select specific materials to compensate the temperature variation of the

The longitudinally coupled IDT shown in **Figure 8(a)** is probably the most used, but it requires reflectors to improve coupling or in other cases attenuators to reduce spurious wave propa‐

piezoelectric layer and to minimise the characteristics of the shift [15].

selectivity of the filter and the more the bandwidth is reduced.

**Figure 6.** Basic SAW filter.

208 Piezoelectric Materials

**Figure 7.** IDT specifications.

*2.1.2. Topologies*

A piezoelectric plate surrounded by metal electrodes is excited in almost all structures in thickness resonance mode to realise a BAW structure. Different arrangements can be done using this basic plate. The principal problem is to keep the properties of the piezoelectric layer as the quality factor. The main constraint is the control of the thickness of the piezoelectric layer as it defines the resonance frequency of the final structure. The piezoelectric layer must be placed on a substrate. The substrate has a significant role in ensuring a high quality factor. It must have a low acoustic impedance compared with the piezoelectric layer. Two main possibilities are used to reach this goal: The first structure, FBAR, is realised on a membrane below which an air cavity is created by micro‐machining to ensure a good acoustic reflection. The main drawbacks of FBAR are the complex process to realise the air cavity and the fragility of the devices. The second structure is a solidly mounted resonator (SMR) where the piezo‐ electric layer and its electrodes are placed on a Bragg acoustic reflector. For both structures, a final layer is deposited at the top and its thickness is adjusted to reach the right resonant frequency. These filters have been highly improved compared with SAW filters, but they remain sensitive to temperature variations and to high power [18].

#### *2.2.2. Film bulk acoustic resonator (FBAR)*

A schematic presentation of an FBAR cross‐section is given in **Figure 9**. The device is composed of a piezoelectric thin film sandwiched between two metallic electrodes, and the bottom one is deposited on an acoustic isolation. The top electrodes can be apodised to minimise spurious modes. A good acoustic isolation is realised by an air cavity underneath the membrane released by bulk micromachining [19, 20].

**Figure 9.** FBAR structure.

*2.2.3. Solidly mounted resonator (SMR)*

For a SMR, the air cavity is replaced by a Bragg mirror as shown in **Figure 10**.

The transpose of a method widely used in optics, which is the Bragg mirror, consists of producing alternating stacks of materials of quarter‐wave layers havingwith low and high acoustic impedances under the active part of the resonator. This principle allows to obtain reflected waves in‐phase with the incident waves generated by the piezoelectric plate to ensure a high quality factor of the structure. The advantages of this structure relative to FBAR are the mechanical strength and the manufacturing process, which is simpler. This structure, however, requires the deposition of additional layers for the Bragg mirror.

**Figure 10.** SMR structure.

#### *2.2.4. Filter topologies*

modes. A good acoustic isolation is realised by an air cavity underneath the membrane released

For a SMR, the air cavity is replaced by a Bragg mirror as shown in **Figure 10**.

requires the deposition of additional layers for the Bragg mirror.

The transpose of a method widely used in optics, which is the Bragg mirror, consists of producing alternating stacks of materials of quarter‐wave layers havingwith low and high acoustic impedances under the active part of the resonator. This principle allows to obtain reflected waves in‐phase with the incident waves generated by the piezoelectric plate to ensure a high quality factor of the structure. The advantages of this structure relative to FBAR are the mechanical strength and the manufacturing process, which is simpler. This structure, however,

by bulk micromachining [19, 20].

210 Piezoelectric Materials

**Figure 9.** FBAR structure.

**Figure 10.** SMR structure.

*2.2.3. Solidly mounted resonator (SMR)*

BAW resonators (FBAR or SMR) are arranged side by side to realise a filter, which simplifies the manufacturing process insofar as a single piezoelectric layer is necessary. However, this category of BAW filters does not allow mode conversion, from asymmetric mode to differential mode, or impedance transformation. Two main architectures of filters exist, ladder filters "Π" or "T" as shown in **Figure 11(a)** and lattice as shown in **Figure 11(b)**. It is necessary to have two types of resonators having different resonant frequencies to build these filters (**Fig‐ ure 12**). The frequency shift can be obtained by adding an additional layer on top of the standard resonator to lower the frequency or by etching the top layer of the stack to increase the frequency. This frequency shift is of great importance in the manufacture of such filters since it determines the width of bandwidth and the level of insertion loss. In **Figure 12**, the transmission coefficient of a ladder structure is presented with the impedances of both resonators. The resonance of the series resonator gives the lower limit of the passband, while the resonance of the shunt resonator gives the upper limit of the passband.

**Figure 11.** Topologies: (a) ladder; (b) lattice; (c) mixed.

**Figure 12.** Transmission coefficients of a ladder structure with two resonators.

The ladder network has a common ground between the inlet and the outlet. The disadvantage of lattice filters is their poor transition band (**Figure 13**) in comparison with ladder filters and with the same number of resonators. It is possible to combine both architectures in a mixed ladder‐lattice architecture to combine the advantages of both topologies [21] (**Figure 11(c)**), allowing for the integration of the performance of selectivity and out‐of‐band rejection of the two topologies.

**Figure 13.** Transmission coefficients of a ladder and ladder filters.

#### **2.3. Coupled resonator filter (CRF)**

The CRF was proposed by Lakin [22] in 2002. This filter is based on three resonators acoustically coupled as shown in **Figure 14**. The input resonator is excited in thickness mode. The acoustic wave will propagate toward the lower resonator through the three coupling layers, guaran‐ teeing optimal transmission between upper and lower piezoelectric layers. The lower piezo‐ electric layer generates an electric field by the inverse piezoelectric effect, which will be guided to the second stack through the continuous electrode of the lower piezoelectric layer. The lower resonator of the second stack, excited by an electric field, emits an elastic wave that travels upward with the coupling layers. This acoustic wave reaches the upper right resonator to finally generate an electric field and the filter output voltage. This structure exhibits a band‐ width comparable with or higher than SAW. As shown in **Figure 15**, the transmission coeffi‐ cient reaches values lower than −80 dB before and after the passband, whereas for BAW filters, **Figure 13** shows that the transmission coefficients, after having reached values higher than -80 dB at the ends of the passband of the filter, increase to values between −40 and −10 dB. The out‐of‐band rejection of CRFs is better than that of BAW filters. The fabrication process of this filter is fully compatible with the microelectronic process. It is possible to realise these filters as stand‐alone devices or as integrated components. With this type of filters, it is possible to realise impedance matching by changing the dimensions of the input resonator or output resonator, except the thickness. A balanced‐unbalanced transformation [23], which is not possible with ladder or lattice structures, can also be designed. Finally, it occupies the smallest surface as compared with other acoustic filters.

**Figure 14.** CRF structure.

The ladder network has a common ground between the inlet and the outlet. The disadvantage of lattice filters is their poor transition band (**Figure 13**) in comparison with ladder filters and with the same number of resonators. It is possible to combine both architectures in a mixed ladder‐lattice architecture to combine the advantages of both topologies [21] (**Figure 11(c)**), allowing for the integration of the performance of selectivity and out‐of‐band rejection of the

The CRF was proposed by Lakin [22] in 2002. This filter is based on three resonators acoustically coupled as shown in **Figure 14**. The input resonator is excited in thickness mode. The acoustic wave will propagate toward the lower resonator through the three coupling layers, guaran‐ teeing optimal transmission between upper and lower piezoelectric layers. The lower piezo‐ electric layer generates an electric field by the inverse piezoelectric effect, which will be guided to the second stack through the continuous electrode of the lower piezoelectric layer. The lower resonator of the second stack, excited by an electric field, emits an elastic wave that travels upward with the coupling layers. This acoustic wave reaches the upper right resonator to finally generate an electric field and the filter output voltage. This structure exhibits a band‐ width comparable with or higher than SAW. As shown in **Figure 15**, the transmission coeffi‐ cient reaches values lower than −80 dB before and after the passband, whereas for BAW filters, **Figure 13** shows that the transmission coefficients, after having reached values higher than -80 dB at the ends of the passband of the filter, increase to values between −40 and −10 dB. The out‐of‐band rejection of CRFs is better than that of BAW filters. The fabrication process of this filter is fully compatible with the microelectronic process. It is possible to realise these filters as stand‐alone devices or as integrated components. With this type of filters, it is possible to realise impedance matching by changing the dimensions of the input resonator or output resonator, except the thickness. A balanced‐unbalanced transformation [23], which is not

two topologies.

212 Piezoelectric Materials

**Figure 13.** Transmission coefficients of a ladder and ladder filters.

**2.3. Coupled resonator filter (CRF)**

**Figure 15.** Simulated transmission coefficients of CRFs.

#### **2.4. Modelling of piezoelectric filter**

Modelling of piezoelectric devices can be achieved in different ways. It depends mainly on the wanted result. The first option is to use a finite element software that is based on 3D modelling. Over the past few years, some commercial software programs were proposed. In general, they are more oriented toward mechanical behaviour analysis, giving information about resonance

modes, strain, displacement and other mechanical information. They are more or less difficult to use according to the machine interface. Most of them can give excellent results on the condition that the user will control, for example, boundary conditions and other features. Another option is to use 1D models, such as the Butterworth‐Van Dyke (BVD) circuit, KLM [24], Mason's model and the transmission line model [25]. Its main advantage is that these models can give information about the electrical behaviour of piezoelectric materials. This is important when the piezoelectric plate is included in an electronic circuit, such as for tele‐ communications transceivers. The main drawback of this option is that 1D models can only be powerful in one dimension and consequently for one vibration mode of the piezoelectric plate. Each model has advantages and drawbacks. The basic BVD model (**Figure 16**) is based on an electric circuit with two parallel branches: one for the electrical behaviour and one for the mechanical behaviour. Its implementation in a circuit simulator is very fast and can give good results if the piezoelectric plate is used at a frequency close to its resonant frequency. If the behaviour of the piezoelectric plate must be analysed far from the resonant frequency, it is possible to add parallel branches to take into account higher resonances. However, the main drawback is that the electrical components of this model do not represent propagation of acoustic waves. Mason's model was proposed in the 1950s and is based on a representation of the piezoelectric plate by a three‐port electrical circuit, taking into account at the same time the electrical behaviour and the propagation of acoustic waves in one dimension. It requires selection of the right vibration mode in order to use the right electrical circuit. The main drawback of Mason's model is that electrical and mechanical losses are not taken into account. For this purpose, the transmission line model based on Mason's model takes into account all losses in the piezoelectric plate. The model for the thickness vibration mode of a piezoelectric plate is represented in **Figure 17(a)**. Furthermore, it is possible to model all materials by a two‐ port network as shown in **Figure 17(b)**. The transmission line matrix model can also be used for SAW transducers [26].

**Figure 16.** BVD model with additional motional branch.

**Figure 17.** (a) Model of piezoelectric plate in thickness mode. (b) Model of non‐piezoelectric layers.

In **Figure 17(a)**, *C*0 is the static capacitor obtained by the following plane capacitor formula:

$$C\_o = \frac{\varepsilon\_{33}(1 - j \tan \delta\_s) S\_p}{t\_p} \tag{1}$$

The impedances *X*1p and *X*2p are given by the following equations:

modes, strain, displacement and other mechanical information. They are more or less difficult to use according to the machine interface. Most of them can give excellent results on the condition that the user will control, for example, boundary conditions and other features. Another option is to use 1D models, such as the Butterworth‐Van Dyke (BVD) circuit, KLM [24], Mason's model and the transmission line model [25]. Its main advantage is that these models can give information about the electrical behaviour of piezoelectric materials. This is important when the piezoelectric plate is included in an electronic circuit, such as for tele‐ communications transceivers. The main drawback of this option is that 1D models can only be powerful in one dimension and consequently for one vibration mode of the piezoelectric plate. Each model has advantages and drawbacks. The basic BVD model (**Figure 16**) is based on an electric circuit with two parallel branches: one for the electrical behaviour and one for the mechanical behaviour. Its implementation in a circuit simulator is very fast and can give good results if the piezoelectric plate is used at a frequency close to its resonant frequency. If the behaviour of the piezoelectric plate must be analysed far from the resonant frequency, it is possible to add parallel branches to take into account higher resonances. However, the main drawback is that the electrical components of this model do not represent propagation of acoustic waves. Mason's model was proposed in the 1950s and is based on a representation of the piezoelectric plate by a three‐port electrical circuit, taking into account at the same time the electrical behaviour and the propagation of acoustic waves in one dimension. It requires selection of the right vibration mode in order to use the right electrical circuit. The main drawback of Mason's model is that electrical and mechanical losses are not taken into account. For this purpose, the transmission line model based on Mason's model takes into account all losses in the piezoelectric plate. The model for the thickness vibration mode of a piezoelectric plate is represented in **Figure 17(a)**. Furthermore, it is possible to model all materials by a two‐ port network as shown in **Figure 17(b)**. The transmission line matrix model can also be used

for SAW transducers [26].

214 Piezoelectric Materials

**Figure 16.** BVD model with additional motional branch.

$$X\_{1\rho} = jZ\_{\rho}S\_{\rho}\tan\left(\frac{\alpha t}{2\nu\_{\rho}}\right) \text{and } X\_{2\rho} = \frac{Z\_{\rho}S\_{\rho}}{j\sin\left(\frac{\alpha t}{\nu\_{\rho}}\right)}\tag{2}$$

where *Z*p, *S*p, *v*p and *t* are the acoustic impedance, surface, longitudinal acoustic wave velocity and thickness, respectively, of the piezoelectric layer. With *Z*<sup>p</sup> =*ρ*p*V*p or *Z*<sup>p</sup> = *<sup>c</sup>* <sup>33</sup>(1 + *<sup>j</sup>*tan*<sup>δ</sup>* m) / *<sup>ρ</sup>*p, the acoustic impedance of the piezoelectric material for longitudinal waves and *V*p is the longitudinal wave velocity in the AlN. The turn ratio of the transformer *N* is given by the following relationship, where *e*33 is the piezoelectric constant:

$$N = \frac{e\_{33} S\_p}{t\_p} \tag{3}$$

Non‐piezoelectric materials (**Figure 17(b)**) are represented as a transmission line with the elastic impedances *X*1m and *X*2m corresponding to the following:

$$X\_{1m} = jZ\_m S\_m \tan\frac{\alpha t\_m}{2V\_m} \quad \text{and} \ X\_{2m} = \frac{Z\_m S\_m}{j\sin\frac{\alpha t\_m}{V\_m}}\tag{4}$$

with *Z*<sup>m</sup> =*ρ*m*V*m being the acoustic impedance of each layer, *V*m the longitudinal wave velocity, *t*<sup>m</sup> the thickness and *ρ*<sup>m</sup> the density. Mechanical losses are considered in each material in the expression of *V*m, which is proportional to the complex elastic constant of the considered material. The surface *S*m involved in the model is equal to the active surface of the piezoelectric plate *S*p.

The last interesting feature of electrical models is that they can be implemented in electric and RF simulators, such as ADS from Keysight. It allows for use of all possible simulations available in this class of simulators, such as *S* parameters, which allow electromagnetic wave propaga‐ tion and electrical impedance matching with input and output ports of the filter to be taken into account, harmonic balance simulation or transient simulation.
