Advances in Piezoelectric Two-Dimensional Materials for Energy Harvesting

*Rafael Vargas-Bernal*

## **Abstract**

The design of piezoelectric energy harvesting systems can be exploited for the development of self-powered sensors, human-powered devices, and regenerative actuators, as well as the development of self-sustained systems with renewable resources. With the introduction of two-dimensional materials, it is possible to implement piezoelectric nanostructures to exploit environmental energies, taking advantage of their flexible mechanical structures. This chapter aims to study the relevant contribution that piezoelectric two-dimensional materials have in energy harvesting. Among the two-dimensional piezoelectric materials analyzed are phosphorene, MXenes, Janus structures, heterostructured materials, and transition metal dichalcogenides (TMDs). These materials are studied through their performance from a piezoelectric point of view. The performance achieved by two-dimensional piezoelectric materials is comparable to or even better than that achieved by bulk piezoelectric materials. Despite the advances achieved so far, many more materials, as well as structures for the implementation of energy harvesting devices or systems, will be proposed in this century, so this research topic will continue to be interesting for research groups around the world.

**Keywords:** two-dimensional materials, piezoelectric materials, transition metal dichalcogenides (TMDs), heterostructures, energy harvesting, morphotropic phase boundary (MPB), Janus structures, MXenes, phosphorene

## **1. Introduction**

The direct piezoelectric effect was discovered in 1880 by Pierre and Jacques Curie in quartz crystal (silicon dioxide, SiO2) [1]. In the direct piezoelectric effect, it is possible to generate an electrical charge proportional to the applied mechanical effort. In the reverse piezoelectric effect, a proportional geometric deformation is achieved by an applied voltage [1]. It was not until 1947 that Shepard Roberts that the first polycrystalline piezoelectric ceramic based on barium titanate (BaTiO3) was discovered to exhibit 100 times more piezoelectricity than quartz. In the 1950s, it was discovered that other oxides such as lead titanate (PbTiO3) and lead zirconate (PbZrO3) have twice the piezoelectric properties of barium titanate. Each material has a Curie

temperature above which piezoelectricity disappears. In addition, for each polycrystalline piezoelectric material, there is a cation ratio that must be optimized to reach the morphotropic phase boundary (MPB), which produces the presence of rhombohedral and tetragonal phases that allow adjusting the piezoelectric properties. In a piezoelectric material, through the applied mechanical stress, it is possible to achieve a total separation of positive and negative charges thanks to the non-centrosymmetric structure of the piezoelectric material. Due to the harmful influence of lead found in piezoelectric materials on the safety of workers who process these materials, as well as the damage to soil and water, researchers are investigating the possibility of developing lead-free piezoelectric materials. Piezoelectric materials are mainly applied to energy harvesting and sensing [2–36]. Without a doubt, one of the contributions that motivated energy harvesting was the use of zinc oxide (ZnO) nanowires as a piezoelectric nanogenerator using nanomaterials [26]. Through the use of two-dimensional materials, it is possible to exploit both semiconductor properties and piezoelectric properties for the tuning and transport of charge carriers [2–36].

The electromechanical interaction in these materials is related to a linear constant between the displacement or electric field and the stress or strain achieved mechanically called piezoelectric coefficient and expressed as C/N or m/V [1]. The piezoelectric coefficients of materials are not uniform along the crystalline directions, that is, are anisotropic, so it is necessary to consider the crystalline direction that will be used in the design of devices that take advantage of piezoelectric properties. Piezoelectric materials enable the development of applications such as energy harvesting in living environments by creating self-powered sources to power electronic devices at the nanoscale. To assess the piezoelectric performance of a material, four main piezoelectric coefficients can be distinguished: the piezoelectric charge coefficient or piezoelectric strain coefficient (*d*), the piezoelectric voltage coefficient or voltage output constant (*g*), the piezoelectric stress coefficient (*e*), and the piezoelectric stiffness coefficient (*h*). Of the 32 possible crystalline solid groups, there are 20 piezoelectric groups and 10 ferroelectric groups. All ferroelectric materials have pyroelectric behavior and all of them are piezoelectric; by contrast, piezoelectric materials are not necessarily pyroelectric or ferroelectric. The most frequently used piezoelectric coefficient is *d* and it relates the open circuit charge density to the applied mechanical stress and regularly uses two subscripts. The first subscript states the direction in which the electrical polarization is induced, and the second subscript specifies the direction in which the stress is applied. This relates the electromechanical coupling coefficient *k*, to the relative dielectric constant *k*<sup>T</sup> at a constant stress, and the elastic compliance at a constant electric field *s* E , as well as the electric permittivity of the vacuum *ε*0, and this is expressed mathematically = ε <sup>0</sup> . *T E asd k k s* The two most important *d* constants in the case of nanomaterials are *d*31 (induced strain in direction 1 per unit electric field applied in direction 3) and *d*11 (induced strain in direction 1 per unit electric field applied in direction 1). The piezoelectric stress coefficient (*e*) can also be defined for the subscripts 11 and 31 of the material for the reverse piezoelectric behavior. For two-dimensional materials, subscripts 11 are considered the in-plane piezoelectric response, while subscripts 31 and 33 are used for the out-ofplane piezoelectric response.

The rest of this chapter has been organized as follows: In Section 2, a summary of the basic concepts associated with piezoelectricity and two-dimensional materials is presented. Next, novel piezoelectric materials are examined using comparative tables and graphs of their in-plane and out-of-plane performance in Section 3. In Section 4, applications of two-dimensional materials in energy harvesting and related topics

are described. Future research directions related to these materials are discussed in Section 5. Finally, the conclusions of this chapter are presented.

## **2. Basic concepts**

A two-dimensional material is a crystalline solid with at least one of its dimensions on the nanoscale. In 2004, graphene was the first two-dimensional material to be discovered, and this is a carbon-based material with a thickness of one atom [2]. Two-dimensional layered materials can be classified as homoatomic when they only contain one chemical element in their structure, such as graphene, phosphorene, antimonene, and heteroatomic when they contain more than 1 chemical element in their structure as in the case of hexagonal boron nitride (*h*-BN), *α*-phase indium sulfide (*α*-In2S3), gallium arsenide (GaAs), etc. Two-dimensional layered materials with homostructure are naturally centrosymmetric because they do not show outof-plane piezoelectricity and have in-plane piezoelectricity; however, this quality depends on the degree of ionization of their covalent bonds. The advantage of two-dimensional layered materials with heterostructure exhibits naturally in-plane and out-of-plane piezoelectricity. Two-dimensional materials are arranged in layers that are stacked and held together through weak van der Waals interactions. Since the discovery of graphene, a large set of two-dimensional materials based on the chemical elements of groups IVA, IIIA-VA, IIB-VIA, and other chalcogenide metals of group B have been discovered and proposed. These materials exhibit unique chemical and physical properties that sometimes differ from those of their bulky counterparts [3, 6–24]. Two-dimensional layered materials are interesting options for implementing energy conversion, harvesting, and storage applications due to their high surface area, processing, and assembly versatility, as well as multiple novel surface chemistries. In these materials, the *d*33 and *g*33 piezoelectric coefficients corresponding to out-of-plane piezoelectricity have zero or none, while the *d*31 and *g*31 piezoelectric coefficients corresponding to in-plane to out-of-plane piezoelectricity have a modest value. Two-dimensional materials to be exploited piezoelectrically must have a large surface-to-volume ratio, extraordinary electrical properties, and extremely low noise level, which also lead them to be used in sensing applications. When comparing sensors based on microelectromechanical systems (MEMS) with sensors based on two-dimensional materials, the latter are much lighter and smaller, have lower power consumption, have more flexibility, and show superior sensitivity. In addition to reducing the thickness of the two-dimensional material, other strategies that can be used to provide these same non-centrosymmetry properties are functionalization and doping of the crystal lattice to favor its anisotropy and thereby achieve better piezoelectric properties [11]. Other approaches that are being used to increase the piezoelectric properties are the generation of defects in the layers or the implementation of heterostructured materials based on two-dimensional materials. Because all the atoms in two-dimensional materials are exposed on their surface, it is possible to tune their physical properties. Within these properties, piezoelectric properties can be used to maximize energy harvest using micro- and nanoelectromechanical systems. Applications of nanoscale piezoelectric materials are transducers, actuators, and energy harvesters for fields such as piezotronics and nanorobotics.

For a material to present piezoelectric properties, the unit cell must not have a symmetrical center so that an anisotropic dipole moment occurs, which is called dielectric polarization [3]. This phenomenon is observed in crystalline and

**Figure 1.** *The basic principle of piezoelectricity in two-dimensional materials.*

semicrystalline dielectric materials when an electric field is applied. The polarization orients the cations and anions found in the material either partially or completely in the direction of the field, the higher the orientation the better the piezoelectric coefficient of the material, especially when this response is achieved under a continuously oscillating field. Traditional applications of piezoelectric materials are in microbalances, high-resolution mechanical actuators, and quartz oscillators [28].

Wearable nanogenerators to generate power can be implemented using mechanically flexible materials that present piezoelectric properties such as two-dimensional materials [2]. The piezoelectric properties of transition metal dichalcogenides (TMDs) appear only in monolayers and disappear in bilayers. So far, transition metal dichalcogenides (TMDs) with odd layer numbers have piezoelectric properties due to the absence of inversion symmetry. When the two-dimensional material is mechanically bent at both ends, the nanosheet is expanded, which causes the polarized charges to deliver a flow of electrons toward an external charge, as illustrated in **Figure 1**. When the two-dimensional material is mechanically released, the electron flow stops. The periodic stretching and releasing of the two-dimensional material can produce an alternating piezoelectric output. This output will be capable of generating a voltage that can be exploited to harvest energy from a source that can generate an oscillating mechanical stress on the two-dimensional material.

## **3. Novel piezoelectric materials**

In Ref. [4] the authors using a data mining algorithm of more than 50,000 inorganic crystals identified that there are 1173 two-dimensional layered materials. Three hundred twenty-five of these materials have piezoelectric monolayers [4]. They also found that there are 98 loosely bound Van der Waals heterostructures. According to Refs. [3, 6], different types of two-dimensional materials with piezoelectric properties can be distinguished. The first type is made up of dichalcogenides based on conventional and Janus-type transition metals. The second type is based on compounds based on the elements of groups IIA and VIA. The third type is based on compounds based on the elements of groups IIIA and VA. The fourth type is made up of compounds based on conventional and Janus-type group IIIA-VIA elements. The fifth type is made up of compounds based on the elements of the IVA and VIA groups.

### *Advances in Piezoelectric Two-Dimensional Materials for Energy Harvesting DOI: http://dx.doi.org/10.5772/intechopen.113754*

Finally, the sixth type is constituted by the compounds based on two elements of the VA group. To produce a non-centrosymmetric structure in materials, it is necessary to use methodologies to modify the interfacial interaction between the ions or layers of the two-dimensional material, produce atomic adsorption on the surface, and/or introduce different defects that can modify the piezoelectric properties [3].

Transition metal dichalcogenides have the *MX*2 structure based on two different chemical elements, a transition metal *M* and a chalcogen *X* atom [3]. Transition metal dichalcogenides have an ultra-thin thickness, tunable bandgap, and unique mechanical, optical, and electronic properties [5]. Multiple derivatives of 2D materials can be obtained through alloying and van der Waals heterojunctions. The symmetry of two-dimensional materials can be modified through the application of stress or electric field, growth on substrates, or the formation of heterojunctions. Symmetry breaking can be achieved using the Janus structure by modifying chalcogenides based on chemical elements of groups VIA and/or IIIA. The Janus structure is obtained by changing the monolayer *MX*2 by the *MXY* structure where *X*, *Y* = S, Se, and Te, and *X* is different from *Y*. This causes the electrical charge distributions between the *M*-*X* and *M*-*Y* layers to differ because the atomic radii and electronegativities are different for elements *X* and *Y*. Examples of these materials are the stable monolayers MoSSe, WSSe, WSeTe, and WSTe and the unstable monolayers MoSeTe and MoSTe.

Atomic layer substitution or Janus substitution can also be used on group IV monochalcogenides to break their symmetry to enrich optical and electrical properties [6]. Group IV two-dimensional monochalcogenides include the following materials: germanium sulfide (GeS2), germanium selenide (GeSe2), germanium telluride (GeTe2), tin sulfide (SnS2), tin selenide (SnSe2), and tin telluride (SnTe2). Due to the geometry of the parent materials, Janus substitution can be performed in three different ways: (1) A chalcogen (S, Se, or Te) is replaced to produce a ternary material, (2) a crystallogen (Ge) is replaced or Sn to produce a ternary material, and (3) both a chalcogen and a crystallogen are replaced to produce a quaternary material. With Janus substitution, 15 two-dimensional materials of group IV monochalcogenides can be produced: Ge2SSe, Ge2STe, Ge2SeTe, Sn2SSe, Sn2STe, Sn2SeTe, GeSnS2, GeSnSe2, GeSnTe2, GeS/SnSe, GeS/SnTe, GeSe/SnS, GeSe/SnTe, GeTe/SnS, and GeTe/SnSe. These materials present mechanical stability, dynamic stability, and energetic stability. Only GeS/SnTe provides dynamic instability. These materials exhibit high piezoelectric coefficients, direct-to-indirect band transitions, as well as high figures of merit for thermoelectric effects. The symmetry breaking produced by Janus substitution leads to high vertical piezoelectric coefficients to increase the efficiency of energy harvesting and other applications that will be described in the chapter.

In the case of transition metal dichalcogenides, the *d*11 piezoelectric coefficient ranges from 2.12 to 13.54 pm/V [3]. In the case of Janus-type transition metal dichalcogenides, the range of the in-plane *d*11 piezoelectric coefficient is from 2.26 to 5.30 mp/V, and for the out-of-plane *d*31 piezoelectric coefficient is from 0.007 to 0.30 pm/V. The *d*11 piezoelectric coefficient for the IIA-VIA compounds ranges from −1.16 to 26.7 pm/V, and this value increases as the atomic number of the group IIA ion decreases and it decreases as the atomic number of the group VIA ion increases. In the case of IIIA-VA compounds with in-plane structure, the *d*11 piezoelectric coefficient ranges from 0.09 to 5.5 pm/V. For these same materials with a non-planar structure, the *d*11 piezoelectric coefficient ranges from 0.02 to 1.50 pm/V and the *d*31 piezoelectric coefficient ranges from 0.02 to 0.57 pm/V. The values of both coefficients increase with decreasing the atomic number of group VA atoms or increasing the atomic number of the group IIIA atom except for aluminum, gallium, and indium nitrides.

For the IIIA-VIA compounds, the values of the *d*11 piezoelectric coefficient are in the range of 1.12 to 1.98 pm/V. The values of the *d*11 and *d*31 piezoelectric coefficients for the IIIA-VIA Janus-type compounds range from 1.91 to 8.47 pm/V and from 0.07 to 0.46 pm/V, respectively. In the case of IVA-VIA compounds, the values of the *d*<sup>11</sup> piezoelectric coefficient range from 75.43 to 250.28 pm/V for a zigzag orientation. For these materials but with armchair orientation, the value of the *d*11 piezoelectric coefficient is in the range of 20.7 to 921.56 pm/V. In the case of a hexagonal orientation, the *d*11 piezoelectric coefficient ranges from −5.65 to −4.63 pm/V. Furthermore, the *e*14 piezoelectric coefficient ranges from 345 to 381 pC/N. Finally, in the VA-VA compounds the alpha (*α*) phase and the beta (*β*) phase occur, and the *d*11 piezoelectric coefficients range from 6.94 to 243.45 pm/V and from 0.67 to 4.83 pm/V, respectively. The values of the piezoelectric coefficients for both phases increase as the atomic number of the VA ion increases.

Two-dimensional piezoelectric nanosheets that have been proposed for energy sensing and harvesting applications include hexagonal boron nitride (*h*-BN), molybdenum sulfide (MoS2), and tungsten diselenide (WSe2) by exploiting their excellent mechanical flexibility and excellent piezoelectric response [10]. The use of twodimensional materials with an odd number of layers allows the inversion of symmetry to be broken, which leads to strong piezoelectricity. For very specific cases, it has been shown that an even number of layers can reach considerable piezoelectric coefficients [11]. Even the armchair or zigzag crystal orientation of the layers of the two-dimensional material used in the piezoelectric material influences the value of the achieved piezoelectric coefficient [12]. In addition to traditional two-dimensional materials, novel emerging materials based on post-transition metals such as tin disulfide (SnS2) are being proposed as piezoelectric materials. These materials present an out-of-plane piezoelectric response of 2 ± 0.22 pm/V for a thickness of 4 nm, whose value is higher than that obtained for materials such as lithium niobate (LiNbO3) whose average value is around 1 pm/V.

The authors in Ref. [7] have estimated through *ab initio* simulation using densityfunctional theory the piezoelectric coefficients of 37 two-dimensional materials among which are transition metal dichalcogenides, oxides, and III-V compounds. The values obtained in Ref. [7] have been plotted through **Figures 2**–**5**, to visualize the wide variety of possible piezoelectric two-dimensional materials. **Figure 2** shows the values reached for the coefficients *d*11 and *e*11 in the plane for two-dimensional transition metal dichalcogenides (TMDs) with a 2*H* phase. The *d*11 coefficient for the dichalcogenides reaches values ranging from 2.12 pm/V for WS2 to 13.45 pm/V for CdTe2. In the case of the *e*11 coefficient for the same materials, it ranges from 184

**Figure 2.** *d11 and e11 piezoelectric coefficients for two-dimensional transition metal dichalcogenides with 2H phase.*

*Advances in Piezoelectric Two-Dimensional Materials for Energy Harvesting DOI: http://dx.doi.org/10.5772/intechopen.113754*

**Figure 3.** *d11 and e11 piezoelectric coefficients for two-dimensional II-VI oxides.*

**Figure 4.**

*d11 and e11 piezoelectric coefficients for two-dimensional III-V compounds.*

**Figure 5.** *d31 and e31 piezoelectric coefficients for two-dimensional III-V compounds.*

pC/N for NbTe2 to 654 pC/N for CrTe2. The values of the piezoelectric coefficients *d*<sup>11</sup> are very similar to those achieved by bulk piezoelectric materials such as quartz-alpha (*α*-SiO2) (2.3 pm/V), gallium nitride (GaN) (3.1 pm/V), and aluminum nitride (AlN) (5.1 pm/V). Most volumetric lead zirconates (PZT) have piezoelectric coefficients *d*<sup>11</sup> on the order of 360 pm/V. Volumetric molybdenum disulfide (MoS2) has a coefficient *e*11 of 290 pC/N.

**Figure 3** depicts the values reached for in the plane coefficients *d*11 and *e*11 in the plane for different two-dimensional II-VI oxides with primitive (*p*) structures. The *d*<sup>11</sup> coefficient for the dichalcogenides reaches values ranging from 1.39 pm/V for BeO to 73.1 pm/V for PbO. In the case of the *e*11 coefficient for the same materials, it ranges from 132 pC/N for BeO to 333 pC/N for CdO.

**Figure 4** illustrates the values reached for in the plane coefficients *d*11 and *e*11 in the plane for some two-dimensional III-V compounds with primitive (*p*) or blende (b) structures. The *d*11 coefficient for dichalcogenides reaches values ranging from 0.02 pm/V for InP to 5.50 pm/V for InN. In the case of the *e*11 coefficient for the same materials, it ranges from 0.5 pC/N for InP to 240 pC/N for BP.

**Figure 5** shows the values reached for the out-of-plane coefficients *d*31 and *e*31 in the plane for different two-dimensional III-V compounds with hexagonal structures. The *d*11 coefficient for dichalcogenides reaches values ranging from 0.016 pm/V for GaSb to 0.568 pm/V for AlAs. In the case of the *e*11 coefficient for the same materials, it ranges from 0.8 pC/N for GaSb to 40.1 pC/N for AlAs.

The authors in Ref. [8] have predicted through first-principles calculations some piezoelectric coefficients of III-V compounds for gallium arsenide (GaAs), gallium phosphide (GaP), gallium antimonide (GaSb), indium arsenide (InAs), indium phosphide (InP), and indium antimonide (InSb). The simulations consider in their determination both clamped ions and relaxed ions [8]. In **Tables 1**–**4**, to study the behavior of piezoelectric two-dimensional III-V compounds, comparisons are made between the values obtained by Refs. [7, 8] for the piezoelectric coefficients *e*11, *e*31, *d*11, and *d*31, respectively. The simulated values illustrate trends related to the assumptions about the theoretical principle and the mechanical physical state in which the ions are simulated to operate, which do not necessarily illustrate the actual experimentally obtained piezoelectric coefficient but serve to confirm the relevance of using two-dimensional materials in energy harvesting compared to other conventional materials already used for years.

The authors of Ref. [9] used density functional perturbation theory (DFPT) and first-principles calculations to predict the piezoelectric coefficients of


#### **Table 1.**

*Compilation of e11 piezoelectric coefficients for some two-dimensional III-V compounds predicted by different simulation techniques.*


#### **Table 2.**

*Compilation of e31 piezoelectric coefficients for some two-dimensional III-V compounds predicted by different simulation techniques.*

### *Advances in Piezoelectric Two-Dimensional Materials for Energy Harvesting DOI: http://dx.doi.org/10.5772/intechopen.113754*


#### **Table 3.**

*Compilation of d11 piezoelectric coefficients for some two-dimensional III-V compounds predicted by different simulation techniques.*


#### **Table 4.**

*Compilation of d31 piezoelectric coefficients for some two-dimensional III-V compounds predicted by different simulation techniques.*


#### **Table 5.**

*Compilation of d11 piezoelectric coefficients for some two-dimensional IIA/IIB-VI oxides predicted by different simulation techniques.*

two-dimensional oxides of IIA/IIB groups. **Tables 5** and **6** present compilations of the values obtained for the *d*11 and *e*11 piezoelectric coefficients. The simulated values vary depending on the simulation conditions and these are very similar considering the second and last columns, but not for all materials. Through the tables and figures in this section, it can be broken down that the values obtained in the simulations must be verified because similar values, or with significant differences, were obtained.

The values are attractive from the point of view that these materials allow the implementation of applications where out-of-plane piezoelectric coefficients can be


#### **Table 6.**

*Compilation of e11 piezoelectric coefficients for some two-dimensional IIA/IIB-VI oxides predicted by different simulation techniques.*

exploited, as opposed to where the materials only exhibit in-plane piezoelectric properties. The computational simulation with different approaches and the experimental corroboration of the piezoelectric performance of two-dimensional materials must be exhaustively studied to guarantee that the values of the piezoelectric coefficients are exploited more efficiently for all the applications that are proposed. In the previous discussion, it can be deduced that some materials do not present piezoelectric properties in bulk size; however, these are present when they are used with thicknesses of some atomic layers and thus are designated two-dimensional materials. Even the opposite of what was mentioned above is also feasible. These materials can be surface-modified appropriately under one or more crystal directions [28]. The addition of atoms and/or defects at the surface level is one of the possibilities to achieve materials with piezoelectric or ferroelectric properties. Even the free electrical charges in these materials must be controlled by either physical or virtual gates depending on the properties of the material to achieve piezoelectricity. The mechanical stability analysis of the piezoelectric behavior has allowed us to determine that the order of stability decreases in the following order: oxides, sulfides, selenides, and tellurides. Furthermore, the mechanical stability is reduced as the radius of the transition metal is decreased in the case of transition metal dichalcogenides (TMDs). To guarantee mechanical stability in a 2D piezoelectric material, it must have a low heat of formation.

A recent type of SnXY Janus monolayers (where X = Te, Se, S, O; Y = Te, Se, S, O; X ≠ Y) is being investigated because it presents static, dynamic, electronic, and thermodynamic stabilities that can be exploited to produce two-dimensional piezoelectric materials [29]. Of this family of two-dimensional materials, those based on tin behave as direct band semiconductors (SnOS and SnOSe with band gaps of 1.74 and 0.33, respectively) or indirect band semiconductors (SnSSe, with band gap of 1.69 eV). The d11 piezoelectric deformation coefficient of selenium tin oxide (SnOSe) reaches a value of 27.3 pm/V, which is an order higher than that reported for materials such as MoS2 or quartz. Two-dimensional tin-based chalcogenides using Janus monolayers can be applied for piezoelectric applications such as energy harvesting and sensors.

## **4. Applications**

The direct use of two-dimensional piezoelectric materials is in the implementation of compact sensors and actuators, flexible electronics, micro-electromechanical

### *Advances in Piezoelectric Two-Dimensional Materials for Energy Harvesting DOI: http://dx.doi.org/10.5772/intechopen.113754*

systems (MEMS), as well as energy harvesting that takes advantage of both the direct and inverse piezoelectric effects [11, 12]. Piezoelectric materials can be applied in the implementation of nanogenerators, information storage, and piezo-catalysis, as well as in biomedicine [3]. With the introduction of sensor networks and the Internet of Things (IoT), batches of sensors that are capable of being self-powered and operating as energy harvesters that exploit piezoelectric properties need to be researched and developed [12]. The first piezoelectric microgenerator was proposed by Glynne-Jones et al. in 2001 [25]. The nanogenerator concept was first proposed by Zhong Ling Wang et al. in 2006 [26]. After this, researchers around the world launched extensive research to develop nanomaterials and nanosystems to convert mechanical energy into electrical energy. Energy harvesters can then be considered the miniaturized replacement for battery-based power supplies for fully portable and/or wearable applications. Piezoelectric nanogenerators provide green and sustainable energy to implement self-powered nanosensors and nanosystems that operate wirelessly and in real time [23]. Nanosystems that directly benefit from self-powered systems are resonators, optoelectronic devices, and biosensors [24]. Furthermore, these nanogenerators can be used in piezo-photonics to tune the performance of photovoltaic devices and/or solar cells. Self-powered systems avoid frequent charging and replacement that are required by battery-based power systems. Battery-based systems take up too much space and are very heavy, which limits their portability and ability to incorporate them into wearable systems. Human activities such as finger typing and breathing may be capable of generating electrical powers on the order of 6.9 mW and 0.83 W, respectively. Wearable electronics require powers of the order of 200 microwatts to 1 watt, which can be achieved by the natural biomechanics of the human being, and its reduction is feasible when designing all systems using nanomaterials. Among the biomechanical movements from which energy can be harvested are the movement of elbow joints, heel strikes, leg movements, and arm swings [26].

If a deformation by tension or compression is applied to a piezoelectric material, a piezo-potential is generated in the pair of metal electrodes found at the ends of the material [3]. Electrons and holes as electric carriers are attracted to the piezopotential with opposite polarity and an electric current is generated on a charge. If the strain is produced continuously, then a continuous current and voltage are generated. In this way, the nanogenerator converts mechanical energy into electrical energy. Nanogenerators based on piezoelectric two-dimensional materials have been implemented using molybdenum diselenide (MoSe2) [12], *α*-phase indium selenide (*α*-In2Se3) [13], black phosphorus or phosphorene (BP) [14], molybdenum disulfide (MoS2) [15], MoS2/In2Se3 Van der Waals heterostructure [16], hexagonal boron nitride (*h*-BN) [10], zinc oxide (ZnO) [17], tungsten diselenide (WSe2) [18], and other materials.

A piezoelectric nanogenerator based on a molybdenum diselenide (MoSe2) nanosheet has been used to power a molybdenum disulfide (MoS2)-based pH sensor and a photodetector based on a molybdenum disulfide-tungsten diselenide (MoS2/ WSe2) [12]. This molybdenum diselenide (MoSe2)-based nanogenerator provides an output voltage of 60 mV with a strain of 0.6%, which is approximately 50% larger than for a molybdenum disulfide (MoS2)-based nanogenerator. Thanks to its excellent performance, this nanogenerator is capable of non-invasively monitoring vital signs to determine the respiratory rate and heart rate.

A simple boron nitride (BN) nanosheet when mechanically deformed can produce an alternating piezoelectric output of 50 mV and 30 pA [10]. For this material, a piezoelectric voltage coefficient (*g*11) was determined experimentally with a value of

2.35 × 10−3 Vm/N. When this material was deposited as active material on a polyimide substrate, an energy harvester was produced with an output voltage of 9 V, a current of 200 nA, and an output power of 0.3 μW.

Human skin is an organ capable of perceiving external environmental stimuli or changes against variables such as temperature, humidity, and pressure [19]. Applications such as prosthetics, medical equipment, wearable devices, robots, and others have benefited from the development of electronic skins. The concept of artificial or electronic skin was proposed in the early 1980s by George Lucas as a future application concept. The first versions implemented showed limited flexibility, low resolution, and poor sensitivity. Therefore, new versions must take advantage of artificial intelligence and wearable technology for the development of health monitoring and prosthetic devices. In addition, the active materials to design these electronic skins must be sensitive, flexible, and independent of their shape and size. The application of piezoelectric nanogenerators and piezotronics allows the implementation of electronic skins that can exceed even the performance of electronic skins for the development of sensors with high spatial resolution, fast response speed, ultra-sensitivity, low power consumption, excellent durability, and ability to electrical self-supply.

IIIA-VIA compounds exhibit the coexistence of in-plane and out-of-plane piezoelectricity caused by hexagonal stacking, which makes them interesting for energy harvesting and electronic skin [13]. An *α*-phase indium selenide (*α*-In2Se3)-based nanogenerator was implemented with 0.76% strain producing a peak voltage and current of 35.7 mV and 47.3 pA, respectively. The *d*33 piezoelectric coefficient of *α*-In2Se3 changes from 0.34 pm/V for a monolayer to 5.6 pm/V for the bulk version.

One of the great challenges of piezoelectric materials is to produce out-of-plane polarization in active materials by exerting stress along the direction perpendicular to the nanosheet [20]. Achieving this polarization could improve the efficiency of piezoelectric transfer and make medical devices such as sphygmomanometers (for indirect blood pressure measurement) and tactical sensors such as bionic robot skins a reality. Heterostructures based on two-dimensional materials such as tin nitride (Sn3N4) indium oxide (In2O3), germanium nitride (Ge3N4)-gallium oxide (Ga2O3), and silicon nitride (Si3N4)-aluminum oxide (Al2O3) have been studied by first-principle calculations seeking to increase the out-of-plane piezoelectric coefficients. The piezoelectric coefficients *d*33 and *e*33 were determined for the two-dimensional materials and their values have been plotted in **Figure 6**. The maximum value of *d*33 is 5290 pm/V corresponding to Sn3N4-In2O3 and the maximum value of *e*33 is 3869 pC/N corresponding to Si3N4-Al2O3. These materials are made prospects for energy harvesting applications as well as for blood pressure meters or bionic skin of robots.

### *Advances in Piezoelectric Two-Dimensional Materials for Energy Harvesting DOI: http://dx.doi.org/10.5772/intechopen.113754*

Phosphorene or black phosphorus (BP) presents interesting properties such as thickness-dependent bandgap and high carrier mobility, due to its anisotropic optical, electronic, mechanical, thermal, and ionic transport characteristics [14]. Phosphorene has a *d*11 piezoelectric strain coefficient of −9.48 pm/V in the armchair direction. Phosphorene nanosheets deliver an output current of 4 pA when a compression strain of −0.72% is applied to them. With all these qualities, phosphorene can be proposed for use in strain sensors, nanogenerators, and piezoelectrically tuned transistors.

Due to a low out-of-plane piezoelectric response of *α*-In2Se3, research has been initiated on the possibility of modifying two-dimensional materials through heterostructures to increase their value. The use of the *α*-In2Se3/MoS2 heterostructure can increase the value of the *d*33 piezoelectric coefficient [16]. The *d*33 piezoelectric coefficient for the *α*-In2Se3/MoS2 heterostructure has a value of 17.5 pm/V. This value increases with the increase in the number of layers of the two-dimensional material but becomes saturated when it reaches 40 nm in thickness.

For a tungsten diselenide (WSe2) bilayer nanosheet, the *d*11 piezoelectric coefficient was 3.26 ± 0.3 pm/V, whose value is higher than the *d*11 coefficient of 2.3 pm/V for volumetric *α*-quartz [18]. The mechanical deformation for a bilayer is 0.95%, while for a monolayer it is 0.63%. WSe2 bilayers feature high piezoelectric coefficients and good mechanical durability for a wide strain range that could be exploited to harvest the energy required by a small liquid crystal display without applying an external power supply.

Next, a diversity of two-dimensional materials is proposed for the development of sensors and energy harvesting, and the values of the piezoelectric coefficients reached are reported, seeking their application, especially for their application outside the plane. Materials such as graphene, thanks to surface modification techniques, can produce in-plane and out-of-plane piezoelectricity reaching 37,000 pC/N for the *d*33 coefficient, 12,500,000 pm/V for the *e*33 coefficient, and 0.3 pm/V for the *d*31 coefficient [21]. For phosphorene, the values of the piezoelectric coefficients are 59 pC/N for the *e*11 coefficient and 1.06 pC/N for the *e*31 coefficient. The piezoelectric coefficient *e*11 for *α*-phase indium selenide (*α*-InSe) is 57 pC/N, and for fluorinated hexagonal boron nitride (F-*h*-BN) it is 84 pC/N. For phosphorene oxide (P4O2)-twodimensional monolayer, the piezoelectric coefficients reach values of 54 pm/V for *d*11, −10 pm/V for *d*31, and 21 pm/V for *d*26. For lithium-doped black phosphor (P4Li2), the piezoelectric coefficient *d*31 ranged from 2.5 to 6.28 pm/V depending on the approximation used. Two-dimensional materials such as antimonene and arsenene can reach strain limits of 58 and 24% in the armchair and zigzag directions, respectively, which are higher when compared to other two-dimensional materials such as graphene, molybdenum disulfide (MoS2), or phosphorene. Other two-dimensional materials that offer out-of-plane piezoelectricity are oxygen-functionalized MXenes (*M*2CO2) where *M* can be scandium (Sc), yttrium (Y), or lanthanum (La), which have a *d*<sup>31</sup> coefficient in the range of 0.4 to 0.78 pm/V, and a coefficient *e*31 in the range of 88 to 196 pC/N, for the case of relaxed ions [22].

The use of vertically aligned two-dimensional flexible zinc oxide nanodiscs for the design of a piezoelectric nanogenerator was reported in Ref. [23]. This nanogenerator used thermally annealed discs and generated a direct current (DC) output voltage of 17 V and a current density of 150 nA/cm<sup>2</sup> . These values increased by 7 times the voltage and 5 times the current density if the pristine version of the same material had been used. This performance improvement was achieved thanks to superficial passivation and the reduction of oxygen vacancies in the two-dimensional material.

A recent alternative that has been reported is the possibility of developing nanogenerators with piezoelectric and triboelectric properties to develop self-powered systems [24]. Achieving the maximum performance of this nanogenerator implies taking advantage of the synergistic coupling between both types of mechanisms leading to increased electrical outputs as well as raising the energy conversion efficiency. When two materials that are electrically charged are placed in friction with each other, electrification is produced which is induced by contact, giving rise to a triboelectric effect. Like the piezoelectric effect, a triboelectric couple is produced which is directly dependent on the relative electrical polarity induced by the induced electrical charge. Two-dimensional materials such as hexagonal boron nitride (*h*-BN), metalorganic scaffolds (MOFs), and transition metal dichalcogenides (TMDs) have been proposed to induce triboelectricity. Some of these materials exhibit both triboelectric properties and piezoelectric effects. Thanks to these qualities, it is possible to decrease the internal resistance, increase the generation of electrical charge, and produce additional electrical charge trapping sites. Involving both electrical mechanisms in the same design makes it feasible to increase the total power output since both effects work efficiently under mechanical deformation, be it compression, tension, and/or vibration. The triboelectric effect operates on the surface of the material, while the piezoelectric effect operates on the volume of the material below the surface. In this way, it is possible to increase polarization by changing the structure of the material, optimizing energy harvesting, and maximizing energy conversion.

According to the previous paragraphs, the main applications derived from energy harvesting with two-dimensional piezoelectric materials are summarized in **Figure 7**. In addition to the applications summarized in **Figure 7**, there is the possibility of developing humidity, magnetic field, and mechanical force sensing. These materials possess unique piezoelectric properties relative to their nanowire-based or volumetric counterparts. Laboratory tests as well as computer simulations have shown that two-dimensional piezoelectric materials can be easily modified to achieve different piezoelectric coefficients by including a different number of layers in the design [27].

An alternative strategic implementation for harvesting mechanical energy is the development of piezoelectric nanogenerators [30]. Materials such as molybdenum disulfide (MoS2) can take advantage of their centrosymmetric structures to produce electricity through the distortion of the crystalline lattice due to the mechanical

**Figure 7.**

*Main applications from energy harvesting with two-dimensional piezoelectric materials.*

### *Advances in Piezoelectric Two-Dimensional Materials for Energy Harvesting DOI: http://dx.doi.org/10.5772/intechopen.113754*

deformation produced by the polarization of the charge of the constituent ions. However, an odd number of layers in the two-dimensional material structure must be used to achieve piezoelectric voltage and current outputs, which is not possible for structures with an even number of layers. Better results in piezoelectric performance are achieved when the number of layers tends to a smaller value.

Recently, a direct current generator using piezoelectric two-dimensional ZnO nanosheets has been implemented to produce an open-circuit voltage of 0.9 V, a current density for short-circuit current of 16.4 μA, and a power density of 600 nW/ cm2 for 4000 cycle operation using 4 kg of force [31]. Due to its mechanical reliability, flexibility, and high output power, this generator has the potential to be used as a power source for portable devices and as a mechanical sensor.

Due to the small thickness and light weight of two-dimensional materials, the suspended application of these materials produces significant mechanical fragility [32]. Therefore, the practical application of these materials involves the use of substrate materials to guarantee good quality of the layers as well as good stability in all the senses previously discussed. In this way, the deformation on the layers of the two-dimensional material can be homogeneous and precisely controlled throughout the substrate. Therefore, the use of polymer-based composite materials that include two-dimensional material is one of the common strategies to replace the use of a fixed and inflexible substrate, especially for wearable piezoelectric electricity generation applications.

Two-dimensional materials based on cobalt telluride (CoTe2) can be used to generate electricity from waste heat using triboelectric and piezoelectric properties as energy harvesters [33]. The piezo-triboelectric nanogenerator can produce a voltage of 5 Volts when a force of 1 N is applied to it operating in a temperature range of 32 to 90 degrees Celsius.

Both mechanical flexibility and bandgap tunability are the strategic advantages of using two-dimensional materials in the implementation of data memories and electronic sensors [34]. To more appropriately exploit two-dimensional materials, it is necessary to increase the quality of their synthesis on a large scale and at a low cost, understand the relationship between the magnetic domain pattern and the applied external electric field, as well as determine the values of the piezoelectric coefficient, Curie temperature, and polarization value and in all crystalline directions of the material.

Piezoelectric materials take advantage of the mechanical deformation produced by ambient energies to develop applications such as nanogenerators as well as optical, mechanical, and magnetic sensors [35]. These make use of electrical polarization to perpendicularly deform a material by either stretching it when a positive voltage is applied or contracting it when a negative voltage is applied. Magnetic behavior is achieved when a two-dimensional material is doped or vacancies or defects are induced on the original structure.

Piezoelectric materials can replace batteries by exploiting energy from the environment for the design of self-powered devices with power consumption in the range of microwatts or milliwatts [36]. These materials offer an environmentally friendly alternative by avoiding the disposal of waste batteries that have not been manufactured and recycled with green technologies.

## **5. Future research directions**

One of the great challenges of this century is to exploit the piezoelectric properties for the implementation of functional, sensitive, and innovative electronic devices

[21]. With the miniaturization of electronic devices, it is necessary to develop strategies for the selection of materials that can be exploited for this purpose. There are more than 7000 possible two-dimensional materials that can be modified to achieve optimization of piezoelectric properties. Among the strategies to modify the piezoelectric properties are deformation, atom or Janus substitution, functionalization, and introduction of defects in a premeditated way. Furthermore, it is possible to stack the two-dimensional nanosheets with similar or dissimilar materials to design heterostructured materials whose piezoelectric properties are completely different from those of their components. Since the piezoelectric properties thanks to these strategies can be tuned for the design of pressure sensors, piezotronics, piezo-catalysis, and energy harvesting, researchers around the world will continue to develop scientific research to take full advantage of piezoelectricity in two-dimensional materials. The suitability of the chosen material to exploit piezoelectricity comes from the following factors: difference in electronegativity between the atoms of the unit cell of the material, impact on health and the environment of the material, cost reduction and ease of the synthesis process of the material, as well as additional material properties. These factors must be considered to choose, design, and integrate the best two-dimensional materials to take the design from the laboratory phase to the practical phase for commercial production.

The study of piezoelectric two-dimensional materials is not complete [21]. Both computational modeling and experimental characterization should be further developed to predict in-plane and out-of-plane piezoelectric behaviors more accurately for various possible theoretical and technological possibilities. In addition, it is necessary to establish standards for test protocols, study the triboelectric effects involved, and complement the necessary terminology to be able to study the piezoelectric properties of two-dimensional materials. Despite the progress made in the research of two-dimensional piezoelectric materials, a lot of research must be carried out to understand their piezoelectric behavior because conventional models and theories are not able to explain the effects found in them. The use of density functional theory and molecular dynamics (MD) calculation will continue to be a vital reference source for calculating, optimizing, and predicting the piezoelectric properties of two-dimensional piezoelectric materials. A comprehensive study of the differential charge density, surface electronegativity, difference in atomic radii, anion-cation polarization ratio, effective Born charges, and elastic constants of two-dimensional piezoelectric materials must be developed to exploit the next generation of applications of high-added value.

## **6. Conclusions**

Two-dimensional materials have undoubtedly attracted the interest of researchers around the world not only for their extraordinary properties but also for the innumerable possibilities of technological development and unprecedented scientific research. The wide range of possibilities to produce anisotropy in its piezoelectric properties extends its applications in energy harvesting, tactical sensors, medical devices, and electronic skins. Despite the advances achieved so far, computational modeling and experimental characterization of the piezoelectric properties of two-dimensional materials are still necessary to achieve a complete study of the most suitable materials to take advantage of the properties in conventional and emerging applications. The graphs of the piezoelectric coefficient values presented in this chapter illustrate, in

*Advances in Piezoelectric Two-Dimensional Materials for Energy Harvesting DOI: http://dx.doi.org/10.5772/intechopen.113754*

addition to the great diversity of possible materials, a wide possibility of both in-plane and out-of-plane coefficient values. Concerning the piezoelectric properties of commonly used zinc oxide (ZnO) and gallium nitride (GaN) nanowires, the piezoelectric coefficients of the two-dimensional materials are 2 orders of magnitude larger. 2D piezoelectric materials can withstand very large deformations for their dimensions. In this century, all two-dimensional materials must be synthesized and studied to exploit the piezoelectric properties, and these can be exploited with maximum efficiency by knowing the specific conditions suitable for each material.

## **Acknowledgements**

The author appreciates the support of the University of Guanajuato to develop this research. The author appreciates the support of the researchers who shared their publications to complement this study.

## **Thanks**

The author wants to thank his wife and son for their support and time to edit this book. The author appreciates the support of Tea Jurcic working for IntechOpen as an author service manager.

## **Author details**

Rafael Vargas-Bernal University of Guanajuato, Salamanca, Guanajuato, Mexico

\*Address all correspondence to: r.vargas@ugto.mx

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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## **Chapter 3**

## Transient Crystal Structure of Oscillating Quartz

*Shinobu Aoyagi and Hiroaki Takeda*

## **Abstract**

Piezoelectric quartz oscillators are widely used to provide a stable clock signal for watches and other electric circuits. The electrically induced mechanical vibration of quartz will be caused by ionic displacements of cationic Si and anionic O sublattices against each other. However, the transient and small ionic displacements during the mechanical vibration cannot be observed by usual X-ray structure analysis. The electrically induced mechanical vibration of quartz is resonantly amplified under an alternating electric field with the resonant frequency. We have revealed the amplified lattice strain and ionic displacements in a resonantly vibrating quartz crystal under an alternating electric field by time-resolved X-ray diffraction. The details of the experiment and application of the technique to other piezoelectric oscillators are introduced in this chapter.

**Keywords:** quartz oscillator, piezoelectricity, time-resolved X-ray diffraction, crystal structure analysis, langasite

## **1. Introduction**

Inorganic crystals that exhibit piezoelectricity are currently used in a wide range of industrial applications, such as oscillators, sensors, transducers, and actuators. Piezoelectricity is the property of crystals that generates an electric polarization *P*i = Σj*d*ij*σ*j (*i* = 1–3) proportional to an applied stress *σ*j (*j* = 1–6). *d*ij are the piezoelectric constants that indicate the degree of piezoelectricity. Crystals belonging to 20 of 32 crystal point groups show piezoelectricity. Piezoelectric crystals exhibit also inverse piezoelectricity that generates a strain *s*j = Σi*d*ij*E*i proportional to an applied electric field *E*i.

The most famous and industrially important piezoelectric crystal is quartz (α-SiO2). Quartz is a naturally abundant mineral and can be synthesized artificially by the hydrothermal method. Quartz oscillates mechanically and electrically at a stable frequency, making it widely used in oscillators to provide reference signals for various devices such as quartz watches. However, quartz has the disadvantage that it cannot be used in high-temperature environments because it undergoes a phase transition from the low-temperature α phase to the high-temperature β phase at 573°C.

The authors have been investigating the mechanism of piezoelectricity of quartz and other piezoelectric crystals and synthesizing new piezoelectric crystals such as langasite-type crystals to develop crystals with high functional piezoelectricity

superior to quartz crystals. In this chapter, we introduce transient crystal structures of quartz and langasite-type crystals oscillating under an alternating electric field measured by synchrotron X-ray diffraction [1–3]. Since inverse piezoelectricity is caused by atomic displacements under electric field, measurement of atomic displacements under an electric field is essential to understand the mechanism of piezoelectricity.

Crystal structure analysis based on X-ray diffraction is a powerful tool to measure atomic displacements in crystals, but highly accurate experiments and analysis are required to measure atomic displacements under electric fields. For example, in the case of a quartz crystal with the thickness of 0.1 mm, when a 1 kV potential difference (10 MV/m in an electric field) is applied between the surfaces, the change in the Si−O bond distance (~0.16 nm = 1.6 Å) is estimated to be only about 10<sup>−</sup><sup>6</sup> nm from its piezoelectric constants. This value is two orders of magnitude less than the standard deviations of bond distances (~10<sup>−</sup><sup>4</sup> nm) determined by conventional X-ray crystal structure analysis.

## **2. Time-resolved X-ray diffraction under alternating electric field**

In order to measure small atomic displacements in piezoelectric crystals under an electric field, the authors have developed a new method for structure analysis of piezoelectric crystals that utilizes resonance under an alternating electric field [1]. Piezoelectric crystals vibrate mechanically and electrically at a certain natural frequency when an instantaneous stress or electric field is applied. For example, the natural frequency of a common quartz oscillator used in industry is 32,768 (215) Hz. When an AC electric field with a frequency equal to this natural frequency is applied, the piezoelectric crystal resonates, producing mechanical and electrical vibrations with a particularly large amplitude. We hypothesized that the resonance under an AC electric field could amplify atomic displacements in piezoelectric crystals to a magnitude that could be measured by X-ray crystal structure analysis. The magnitude of the amplification effect in resonance depends on the quality factor (*Q*-value) of the crystal defined by 2*π*[(energy stored in the system)/(energy lost from the system in one period of vibration)]. The larger the *Q*-value, the smaller the energy dissipation and the greater the amplification effect. The *Q*-value of piezoelectric crystals used in resonators is particularly high, exceeding 106 for quartz crystals.

Even if the atomic displacements involved in piezoelectricity can be greatly amplified by resonance under an alternating electric field, it is actually impossible to measure them using conventional X-ray diffraction. Conventional X-ray diffraction measures X-ray diffraction images during X-ray irradiation for several seconds to several minutes while the crystal is rotating, so it is impossible to measure instantaneous X-ray diffraction images of piezoelectric crystals vibrating at frequencies from kHz to MHz. Synchrotron radiation (SR) X-rays with high brilliance and short pulse duration are useful for the measurement of instantaneous X-ray diffraction images. SR is an electromagnetic wave emitted tangentially to the trajectory of an electron bunch accelerated to nearly the speed of light when the trajectory is bent by a magnetic field. We have been conducting SR time-resolved X-ray diffraction experiments under an AC electric field at SPring-8, the large SR facility (Hyogo, Japan), to measure transient atomic displacements in piezoelectric crystals resonating under an AC electric field. The shortest pulse duration is about 50 ps. By repeatedly irradiating a piezoelectric crystal resonating under an AC electric field with highly brilliant and

*Transient Crystal Structure of Oscillating Quartz DOI: http://dx.doi.org/10.5772/intechopen.107414*

**Figure 1.** *Schematic of time-resolved X-ray diffraction under AC electric field.*

short pulse X-rays synchronized with the AC electric field, instantaneous x-ray diffraction images when the atomic displacements reach the maximum can be measured with high accuracy (**Figure 1**).

## **3. Transient atomic displacements in quartz oscillator**

Transient atomic displacements in a quartz oscillator were successfully measured by the SR time-resolved X-ray diffraction under an AC electric field [1]. Quartz crystal belongs to the trigonal crystal system with the point group 32. There are two types of crystal polymorphs in quartz: right and left quartz, which are enantiomorphs of each other. The industrially used quartz crystal is a right crystal with the space group *P*3221. The crystal structure of the right quartz crystal is shown in **Figure 2a**. The crystal structure of quartz consists of corner shared SiO4 tetrahedra with the Si−O bond distance of 1.61 Å, the O−Si−O bond angle of 109 degrees, and the Si−O−Si bond angle of 143 degrees. The direction along the twofold axis is called the *X*-axis, the direction perpendicular to the twofold and threefold axes is called the *Y*-axis, and the direction along the threefold axis is called the *Z*-axis. The *X*, *Y*, and *Z*-axes correspond to the [100], [120], and [001] crystal band axes of the primitive trigonal lattice, respectively. The nonzero piezoelectric constants of the right quartz are *d*11 = −*d*12 = −*d*26/2 = −2.31 pC∙N<sup>−</sup><sup>1</sup> , and *d*14 = −*d*25 = −0.727 pC∙N<sup>−</sup><sup>1</sup> [4]. The right

#### **Figure 2.**

*(a) Crystal structure of quartz viewed along the twofold axis (square box is the trigonal unit cell), and (b) schematic of AT-cut quartz oscillator.*

quartz and left are piezoelectrically distorted under the electric field *E*1 along the *X*-axis and *E*2 along the *Y*-axis, but not under the electric field *E*3 along the *Z*-axis.

A commercially available AT-cut quartz oscillator with an oscillation frequency of 30 MHz was used as the sample for the measurement. AT-cut oscillators are plate-shaped crystal (**Figure 2b**) cut along the plane including the *X*-axis, which is perpendicular to the direction tilted from the *Y*-axis to the *Z*-axis by *δ* = 35 degrees. It is widely used in industry because of its small temperature variation of oscillation frequency near room temperature. When no electric field is applied, the *X*, *Y*, and *Z*-axes are orthogonal to each other, but when an electric field *E* is applied along the direction perpendicular to the AT-cut plane, the shear strain given by *s*5 = *d*25*E*2 and *s*6 = *d*26*E*<sup>2</sup> causes a distortion of the angle between the *X* and *Z*-axes (*β* angle) and angle between the *X* and *Y*-axes (*γ* angle) from 90 degrees (**Figure 2b**). Here, *E*2 = *E*cos*δ*. The angle between the *Y* and *Z*-axes (*α* angle) is not distorted under the electric field. The oscillation frequency of an AT-cut oscillator *f*0 is given by *f*0 = 1664/*h* (MHz) where *h* (μm) is the thickness of the crystal. To reduce X-ray absorption and extinction effects, a thin and high frequency AT-cut quartz oscillator with *h* = 55 μm (*f*0 = 30 MHz) was used as a sample.

X-ray diffraction experiments were performed at the SPring-8 beamline BL02B1 [5] using X-rays with a wavelength of 0.4 Å. A large cylindrical curved imaging plate was used as the X-ray detector. First, measurements under a DC electric field were performed. Piezoelectric distortions of *β* and *γ* angles from 90 degrees under a DC electric field of *E* = 36 MV∙m<sup>−</sup><sup>1</sup> (2.0 kV in potential difference) were −0.002(2) and −0.007(3) degrees, respectively. These values are consistent with the values of −0.001 and −0.008 degrees calculated from the piezoelectric constants. Crystal structures determined from more than 4000 Bragg reflection intensities measured under DC electric fields of *E* = +36 and −36 MV∙m<sup>−</sup><sup>1</sup> were compared. However, no differences in the bond distances and angles exceeding their standard deviations were observed.

Time-resolved X-ray diffraction of a resonantly vibrating AT-cut quartz oscillator (**Figure 1**) was performed by applying a sinusoidal AC electric field with a frequency of 30 MHz and an electric field amplitude of 0.18 MV∙m<sup>−</sup><sup>1</sup> to the sample. Resonance of the sample was confirmed by detecting the current flowing in the circuit with a current probe and displaying it on an oscilloscope. The resonant sample was irradiated with pulsed X-rays with a pulse duration of ~50 ps at a repetition rate of 26 kHz using an X-ray chopper [6]. In order to synchronize the oscillation of the sample and the pulsed X-rays, the ratio of the resonance frequency of the sample to the repetition frequency of the X-rays must be an exact integer ratio. However, both of the resonance frequency of the sample and the repetition frequency of the X-rays cannot be tuned freely. The authors have synchronized the resonance of the sample with pulsed X-rays by modulating the 30 MHz AC electric field at 26 kHz. This method enables instantaneous measurement of X-ray diffraction images of a resonant sample with a time resolution of less than 1 ns. The period of the resonant sample is 33 ns = 1/30 MHz. By varying the delay time Δ*t* of the alternating electric field to the pulsed X-rays from 0 to 33 ns, the time variation in one period of the X-ray diffraction images of the resonant sample was measured.

**Figure 3a** shows time dependences of deviations of *α*, *β*, and *γ* lattice angles from 90 degrees (Δ*α*, Δ*β*, and Δ*γ*) obtained by the least-squares method from the positions of several hundred Bragg reflections measured on the X-ray diffraction images. It can be seen that, during the resonance, *α* angle remains almost unchanged with time, while *β* and *γ* angles oscillate sinusoidally and significantly. The amplitudes of the Δ*β* and Δ*γ*

*Transient Crystal Structure of Oscillating Quartz DOI: http://dx.doi.org/10.5772/intechopen.107414*

**Figure 3.**

*(a) Time variation of Δα, Δβ, and Δγ, and (b) Si−O−Si bond angles of AT-cut quartz crystal under AC electric field.*

oscillations are 0.10 and 0.15 degrees, respectively. These values are the several tens of times larger than the aforementioned values of −0.002(2) and −0.007(3) degrees under the DC electric field of 36 MV∙m<sup>−</sup><sup>1</sup> . Δ*β* and Δ*γ* under a DC electric field of 0.18 MV∙m<sup>−</sup><sup>1</sup> calculated from the piezoelectric constants are −1 × 10<sup>−</sup><sup>5</sup> and −4 × 10<sup>−</sup><sup>5</sup> degrees, respectively. Thus, the resonance under an AC electric field amplifies Δ*β* and Δ*γ* by a factor of 1 × 104 and 4 × 103 , respectively.

The crystal structures at the delay times Δ*t* = 9 and 25 ns when Δ*β* and Δ*γ* reach the negative and positive maxima, respectively, were obtained by the least-squares method from more than 3000 Bragg reflection intensities. Comparison of the two crystal structures shows that the Si−O bond distances and O−Si−O bond angles in the SiO4 tetrahedra do not differ by more than their standard deviations (0.002 Å for the bond distances and 0.1 degrees for the bond angles). However, a clear difference was observed in the Si−O−Si bond angles between the SiO4 tetrahedra. The triclinic crystal structure of the quartz crystal distorted under an electric field consists of three independent SiO4 tetrahedra with six independent oxygen atoms (O(1)~O(6)). The six independent Si−O−Si bond angles at Δ*t* = 9 and 25 ns are shown in **Figure 3b**. The abscissa is the number of independent oxygen atoms 1~6. Among O(1)~O(6), large deformations were observed in the Si−O−Si bond angles, especially around O(2) and O(3).

The large deformation of the Si−O−Si bond angles around O(2) and O(3) can be understood from the displacements of the anionic O atoms under the electric field and the arrangements of the Si−O−Si bonds relative to the electric field. The Si−O bonds have both covalent and ionic characters. The electron density distribution was calculated from the measured X-ray diffraction intensities to estimate the numbers of electrons of each atom. The charge deviations from neutral were +2.8*e* for Si and −1.4*e* for O (*e* is the elementary charge). The Si and O atoms, which are positively and negatively charged, respectively, are expected to be displaced in opposite directions upon application of an electric field. However, each Si atom is covalently bonded to four O atoms in tetrahedral coordination, which makes it extremely difficult to displace the Si atoms by an electric field. O atoms are subject to a large repulsive force due to the Si−O covalent bonds when the displacement direction is parallel to the Si−O−Si plane, but can be displaced relatively easily when the displacement direction is perpendicular to the Si−O−Si plane. Comparing the angles between the electric field and the Si−O−Si planes of O(1)~O(6), O(2), and O(3) with the angles about

#### **Figure 4.**

*Oxygen atomic displacements in AT-cut quartz crystal viewed along the twofold axis under electric field* **E** *perpendicular to the crystal plane.*

66 degrees are the closest to perpendicular. The Si−O−Si bond angles centered on O(2) and O(3) deform with the oxygen displacements in the direction perpendicular to the Si−O−Si planes. The relationship between the crystal structure, electric field, and atomic displacements of O(2) and O(3) is shown in **Figure 4**.

As shown above, the piezoelectric distortion of a quartz crystal under an electric field is caused by the displacement of oxygen ions in the direction perpendicular to the Si−O−Si plane due to the electric field and the accompanying deformation of the Si−O−Si bond angles. Mechanical and electrical vibrations with high *Q*-values of quartz crystals are caused by the restoring force acting on the Si−O−Si bond angles, which causes a harmonic vibration of the oxygen ions with little damping. The combination of resonant vibration under an alternating electric field and SR time-resolved X-ray diffraction has revealed the transient atomic displacements and mechanism of piezoelectricity in quartz. Using this experimental technique, the authors next performed similar transient atomic displacement measurements under an AC electric field on langasite-type crystals, which are useful as piezoelectric materials for hightemperature applications [2, 3].

## **4. Applications to langasite-type crystals**

Langasite (La3Ga5SiO14, LGS) is a piezoelectric crystal belonging to the same crystal point group 32 as quartz and does not show a phase transition until its melting point around 1470°C. Its piezoelectric constants are several times larger than those of quartz, *d*11 = −5.95, *d*14 = 5.38 pC∙N<sup>−</sup><sup>1</sup> [7]. **Figure 5a** shows the crystal structure of langasite. The space group is *P*321, and the crystal structure consists of GaO6 octahedra, GaO4, and Ga1/2Si1/2O4 tetrahedra, and La atoms coordinated by eight oxygen atoms. La atoms can be substituted with other rare earth (RE) elements such as Pr and Nd, and it is known that the piezoelectric constants decrease as the ionic radius of the RE elements decreases [7–10]. In Nd3Ga5SiO14 (NGS), where La is replaced by Nd, *d*11 = −4.05, *d*14 = 2.07 pC∙N<sup>−</sup><sup>1</sup> [7]. Transient atomic displacement measurements under an AC electric field were performed on LGS and NGS to understand the origin

*Transient Crystal Structure of Oscillating Quartz DOI: http://dx.doi.org/10.5772/intechopen.107414*

**Figure 5.**

*(a) Crystal structure of langasite-type crystal viewed along the twofold axis (square box is the trigonal unit cell), and (b) coordination structure around a RE atom.*

of the change in piezoelectric constants due to RE element substitution, in addition to the mechanism of piezoelectricity in langasite-type crystals. The samples used were a commercially available *Y*-cut LGS crystal with an oscillation frequency of 28 MHz (thickness: 0.05 mm) and a homemade *Y*-cut NGS crystal with an oscillation frequency of 13 MHz (thickness: 0.09 mm). X-ray diffraction experiments were performed at SPring-8 BL02B1 using X-rays with a wavelength of 0.3 Å.

The crystal structures of LGS and NGS in the absence of an electric field were analyzed and compared first. There was little difference in the Ga(Si)−O bond distances between them. The RE (La, Nd) atoms are coordinated with eight O atoms, but because the RE atoms are on the twofold *X*-axis, there are only four independent RE−O bonds (**Figure 5b**). Among the four types of RE−O bonds, large changes in bond lengths were observed in the two short RE−O bonds due to elemental substitutions. The bond lengths of these two RE−O bonds are 2.355 and 2.508 Å for LGS and 2.301 and 2.431 Å for NGS, respectively, indicating that the substitution of La with Nd shortened the bond distance by more than 0.05 Å. In accordance with the shortening of the bond distances, the Ga−O−Ga bond angles centered on the oxygen atoms of these two types of RE−O bonds were also significantly deformed. The two Ga−O−Ga bonds are the only Ga−O−Ga bonds existing in the crystal structure, which bridging a GaO6 octahedron and a GaO4 tetrahedron, and a GaO4 tetrahedron and a Ga1/2Si1/2O4 tetrahedron, respectively. The bond angles of these two types of Ga−O−Ga(Si) bonds are 114.2 and 122.3 degrees for LGS and 112.6 and 120.3 degrees for NGS, indicating a decrease in the bond angles of about 2 degrees with the substitution of La with Nd. In addition, NGS has a larger deformation of the GaO6 octahedron from the regular octahedron than LGS. The O−Ga−O bond angles in the GaO6 octahedron are 90 and 180 degrees when the GaO6 octahedron is a regular octahedron. The maximum deviations of the O−Ga−O bond angles from 90 and 180 degrees in the GaO6 octahedron are 12.9 and 13.8 degrees for LGS and 15.2 and 16.2 degrees for NGS.

X-ray diffraction experiments under a DC electric field showed that the lattice deformations of Δ*β* and Δ*γ* were consistent with the piezoelectric constants, as in the case of quartz. Piezoelectric constants estimated from Δ*β* and Δ*γ* were *d*11 = −5.7 and *d*14 = 5.3 pC∙N<sup>−</sup><sup>1</sup> for LGS and *d*11 = −3.8 and *d*14 = 2.3 pC∙N<sup>−</sup><sup>1</sup> for NGS. Time-resolved X-ray diffraction experiments under an AC electric field were performed with a sinusoidal AC electric field applied to the sample at the resonant frequency, as in the case of the quartz crystal. The electric field amplitudes were 0.20 MV∙m<sup>−</sup><sup>1</sup> for LGS and 0.16 MV∙m<sup>−</sup><sup>1</sup> for NGS. The repetition frequency of the pulsed X-rays was 52 kHz for LGS and 69 kHz for NGS. As in the case of the quartz crystal, the time variation of the

**Figure 6.** *Time variations of Δβ and Δγ for the (a) LGS and (b) NGS oscillators resonating under AC electric field.*

X-ray diffraction image of the resonant sample in one period was measured by changing the delay time of the AC electric field to the pulsed X-rays, and the time variation of the lattice constant and crystal structure were investigated from these images.

**Figure 6a** and **b** show the time variation of Δ*β* and Δ*γ* for the LGS and NGS oscillators resonating under an AC electric field. As in the case of the quartz crystal (**Figure 3a**), a large sinusoidal oscillation was observed in Δ*γ*. The amplitudes of the oscillation in Δ*γ* were 0.10 degrees for the LGS and 0.04 degrees for the NGS, respectively. The difference between the two values can be attributed mainly to the difference in piezoelectric constants as described above. The amplification factor of Δ*γ* under an AC electric field due to resonance is more than 5 × 102 from their piezoelectric constants, but smaller than that of a quartz crystal around 1 × 104 . The larger amplification effect in quartz is attributed to its *Q*-value larger than those of LGS and NGS. The amplification effect of Δ*β* due to resonance was observed in the AT-cut quartz crystal (**Figure 3a**), but not in the *Y*-cut LGS and NGS crystals (**Figure 6**). An AC electric field was applied along the direction 35 degrees inclined from the *Y*-axis to the *Z*-axis in the AT-cut quartz crystal (**Figure 2b**), while it was applied along the *Y*-axis in the *Y*-cut LGS and NGS crystals. Δ*β* is generated by the atomic displacements perpendicular to the *Y*-axis, so an electric field component perpendicular to the *Y*-axis will be necessary to amplify Δ*β*.

The crystal structures of LGS and NGS resonating under the AC electric field at the delay times when Δ*γ* reached their negative and positive maxima were analyzed and compared. Changes in RE−O bond distances, Ga−O−Ga, and O−Ga−O bond angles from the negative to positive maxima of Δ*γ* are shown in **Figures 7** and **8**. No changes in the Ga(Si)−O bond distances were observed during resonance for both LGS and NGS. In the triclinic distorted structure with 24 independent RE−O bonds under an electric field, three RE−O bonds in LGS and one RE−O bond in NGS showed a change in their bond distances of more than ±0.01 Å with time (**Figure 7a**). Among the four types of independent RE−O bonds in the trigonal non-distorted structure, only the longest RE−O bonds with the bond distances longer than 2.85 Å showed a clear time variation in the resonance. The two short RE−O bonds, which vary greatly with RE atomic substitution, show no time variation in the resonance. Focusing on the Ga−O−Ga bond angles, we observed that among the 12 types of Ga−O−Ga(Si) bonds in the distorted state under an electric field, four types of bonds *Transient Crystal Structure of Oscillating Quartz DOI: http://dx.doi.org/10.5772/intechopen.107414*

#### **Figure 7.**

*Changes in (a) RE−O lengths and (b) Ga−O−Ga angles under AC electric field for Y-cut LGS and NGS resonators.*

**Figure 8.**

*Changes in O−Ga−O angles in (a) GaO6 octahedra, (b) GaO4, and (c) Ga1/2Si1/2O4 tetrahedra under AC electric field for Y-cut LGS and NGS resonators.*

in the LGS and one type of bond in the NGS showed a change in their bond angles of more than ±0.25 degrees with time (**Figure 7b**). These results indicate that replacing La with Nd significantly reduces the deformation of the RE−O bond distances and the Ga−O−Ga bond angles during resonance as RE−O bond distances are shortened. The decrease in the piezoelectric constants due to the replacement of La with Nd is mainly attributed to the fact that the deformation of the Ga−O−Ga bond angles is prevented due to the shortening of the RE−O bond distances.

Next, the deformation of O−Ga−O bond angles in GaO6 octahedra, GaO4, and Ga1/2Si1/2O4 tetrahedra in the resonance state was investigated. Of the 15 independent O−Ga−O bonds in the GaO6 octahedra of the distorted structure under an electric field, no and 5 O−Ga−O bonds showed a time variation of their bond angles greater than ±0.25 degrees in LGS and NGS, respectively (**Figure 8a**). In the GaO4 tetrahedra, of the 18 independent O−Ga−O bonds, 5 and 5 O−Ga−O bonds showed a time variation greater than ±0.25 degrees in LGS and NGS, respectively (**Figure 8b**). In the Ga1/2Si1/2O4 tetrahedra, of the 12 independent O−Ga−O bonds, 2 and 1 O−Ga−O bonds showed a time variation greater than ±0.25 degrees in LGS and NGS, respectively (**Figure 8c**). The substitution of La with Nd increases the deformation of the GaO6 octahedra from the regular octahedron and facilitates the deformation of the O−Ga−O bond angles of the GaO6 octahedra during resonance. The Ga−O−Ga bond angles in NGS are not easily deformed due to the shortening of RE−O bonds, but the GaO6 octahedra are deformed instead. The deformation of GaO4 and Ga1/2Si1/2O4 tetrahedra during resonance was observed in both LGS and NGS, but no deformation of SiO4 tetrahedra during resonance was observed in quartz. Therefore, the O−Ga−O bond is more flexible than the O−Si−O bond, and as a result, the piezoelectric constants of LGS and NGS are larger than those of quartz. However, the piezoelectric deformation of LGS and NGS is caused by the deformation of multiple bonds with different force constants, resulting in greater energy dissipation and lower amplification effect (*Q*-value) in resonance than those of quartz.

As described above, the mechanism of piezoelectricity and the effect of RE substitution in langasite-type crystals were understood by the combination of resonance phenomena under an AC electric field and SR time-resolved X-ray diffraction. LGS shows larger piezoelectric deformation than quartz due to the deformation of GaO4 and Ga1/2Si1/2O4 tetrahedra. In NGS, the piezoelectric deformation is smaller because the shortening of the RE−O bond distances prevents the deformation of the Ga−O− Ga bond angles, but instead the GaO6 octahedral deformation is observed. These findings will be useful for the design and development of piezoelectric crystals with higher functionality than quartz.

## **5. Summary**

In this chapter, we introduce transient crystal structures of quartz and langasitetype crystals resonating under an AC electric field observed by SR time-resolved X-ray diffraction. The method can detect small transient atomic displacements in piezoelectric crystals by resonantly amplifying them under an alternating electric field by a factor of 103 –104 . We will use this method for structural analyses of other piezoelectric crystals and expand its range of application.

## **Acknowledgements**

This work was supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS) (Grants Nos. JP22H02162, JP19H02797, JP16K05017, and JP26870491), Tatematsu Foundation, Toyoaki Scholarship Foundation, Daiko Foundation, and the Research Equipment Sharing Center at the Nagoya City University. The synchrotron radiation experiments were performed at SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI).

## **Conflict of interest**

The authors declare no conflict of interest.

*Transient Crystal Structure of Oscillating Quartz DOI: http://dx.doi.org/10.5772/intechopen.107414*

## **Author details**

Shinobu Aoyagi1 \* and Hiroaki Takeda2

1 Department of Information and Basic Science, Graduate School of Science, Nagoya City University, Nagoya, Japan

2 Faculty of Engineering, Department of Applied Chemistry, Saitama University, Saitama, Japan

\*Address all correspondence to: aoyagi@nsc.nagoya-cu.ac.jp

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Aoyagi S, Osawa H, Sugimoto K, Fujiwara A, Takeda S, Moriyoshi C, et al. Atomic motion of resonantly vibrating quartz crystal visualized by timeresolved X-ray diffraction. Applied Physics Letters. 2015;**107**:201905. DOI: 10.1063/1.4935591

[2] Aoyagi S, Osawa H, Sugimoto K, Takeda S, Moriyoshi C, Kuroiwa Y. Time-resolved crystal structure analysis of resonantly vibrating langasite oscillator. Japanese Journal of Applied Physics. 2016;**55**:10TC05. DOI: 10.7567/ JJAP.55.10TC05

[3] Aoyagi S, Osawa H, Sugimoto K, Nakahira Y, Moriyoshi C, Kuroiwa Y, et al. Time-resolved structure analysis of piezoelectric crystals by X-ray diffraction under alternating electric field. Japanese Journal of Applied Physics. 2018;**57**:11UB06. DOI: 10.7567/ JJAP.57.11UB06

[4] Bechmann R. Elastic and piezoelectric constants of alpha-quartz. Physical Review. 1958;**110**:1060-1061. DOI: 10.1103/PhysRev.110.1060

[5] Sugimoto K, Ohsumi H, Aoyagi S, Nishibori E, Moriyoshi C, Kuroiwa Y, et al. Extremely high resolution single crystal diffractometory for orbital resolution using high energy synchrotron radiation at SPring-8. AIP Conference Proceedings. 2010;**1234**:887-890. DOI: 10.1063/1.3463359

[6] Osawa H, Kudo T, Kimura S. Development of high-repetition-rate X-ray chopper system for time-resolved measurements with synchrotron radiation. Japanese Journal of Applied Physics. 2017;**56**:048001. DOI: 10.7567/ JJAP.56.048001

[7] Takeda H, Izukawa S, Shimizu H, Nishida T, Okamura S, Shiosaki T. Growth, structure and piezoelectric properties of Ln3Ga5SiO14 (Ln = La, Nd) single crystals. Transactions of the Materials Research Society of Japan. 2005;**30**:63-66

[8] Sato J, Takeda H, Morikoshi H, Shimamura K, Rudolph P, Fukuda T. Czochralski growth of RE3Ga5SiO14 (RE = La, Pr, Nd) single crystals for the analysis of the influence of rare earth substitution on piezoelectricity. Journal of Crystal Growth. 1998;**191**:746-753. DOI: 10.1016/ S0022-0248(98)00362-5

[9] Iwataki T, Ohsato H, Tanaka K, Morikoshi H, Sato J, Kawasaki K. Mechanism of the piezoelectricity of langasite based on the crystal structures. Journal of the European Ceramic Society. 2001;**21**:1409-1412. DOI: 10.1016/ S0955-2219(01)00029-2

[10] Araki N, Ohsato H, Kakimoto K, Kuribayashi T, Kudoh Y, Morikoshi H. Origin of piezoelectricity for langasite A3Ga5SiO14 (A = La and Nd) under high pressure. Journal of the European Ceramic Society. 2007;**27**:4099-4102. DOI: 10.1016/j.jeurceramsoc.2007.02.177

## **Chapter 4**

## Magnetoelectric Composites-Based Energy Harvesters

*Tarun Garg and Lickmichand M. Goyal*

## **Abstract**

Electrical energy generation from renewable resources has been a quest in the last few decades to meet the energy demand of electrical appliances and gadgets. More importantly, portable gadgets and devices, wireless sensors, etc., which rely on batteries require intermittent charging, and it is difficult to find an omnipresent continuous electrical energy source connected to a power station for these batteries. Alternate to these power stations connected to electrical energy sources is harvesting the energy from omnipresent mechanical and acoustic vibrations and AC magnetic field. Energy harvesting from these waste energy resources is possible using piezoelectric and magnetoelectric materials. This chapter would discuss in detail various mechanisms and stimuli, which may be synergistically used to harvest energy from piezoelectric materials-based energy harvesters.

**Keywords:** magnetoelectric, multiferroic, energy harvesting, piezoelectric, ferromagnetic

## **1. Introduction**

Harvesting electrical energy from various wasted forms of energy in the environment could be a way to develop sustainable energy sources. However, to do so, we need to develop smart materials and structures, which could convert various wasted forms of energy into electrical energy. These smart materials and structures may enable us to develop sustainable energy sources required for powering up low-power electronic devices [1, 2]. In the recent past, a lot of research attention has been devoted to converting omnipresent mechanical and acoustic vibration energy to electrical energy using electromagnetic [3–5], electrostatic [6–8], and piezoelectric [9–11] transductions. The most popular technique among the methods is energy harvesting using the piezoelectric effect, which is proved to be advantageous over other methods due to its easier execution and higher power density. These piezoelectric energy harvesters are capable of harvesting energy from various kinds of vibration sources involving translational and rotary motions [12–14], wind [15, 16], or some fluid flow induced vibrations [17, 18]. Another ubiquitous form of wasted energy is magnetic energy found around the electrical current-carrying wires and appliances. This energy can be harvested by using magnetoelectric (ME) composites. An ME composite comprises a ferromagnetic (FM) material and a piezoelectric material combined with various phase connectivity schemes. On application of a magnetic field across such a composite, FM material

undergoes magnetostriction, which leads to an elastic strain in the material. This elastic strain is transferred to the connected piezoelectric phase *via* a mechanical coupling. Due to this strain in the piezoelectric material, an electric field is generated in a transverse direction across the piezoelectric phase. This electric field is harvested in the form of electric energy using these ME composites. In other words, the application of a magnetic field across these composites produces an electric field. This effect is termed as the direct magnetoelectric effect. The converse effect is also true in these ME composites, where the application of an electric field leads to the generation of a magnetic field. Both effects have been explored in numerous applications such as energy harvesting, magnetoelectric transformers, AC and DC magnetic field sensors, phase shifters, and resonators. [19]. However, in this chapter, we will be discussing the direct magnetoelectric effect in ME composites, which is useful for energy harvesting. This chapter is organized in the following way: We will begin with a discussion of the magnetoelectric effect in magnetoelectric composites, the materials requirements, and the properties of the ME effect. A list of candidate materials for ME composites has been provided to give the idea of specific materials to the readers. Then, applications of these ME composites, specifically energy harvesting, have been discussed.

## **2. Magnetoelectric effect**

ME effect is the induction of electric polarization in a material under an applied external magnetic field or conversely, the generation of magnetization due to an applied external electric field [20]. This induced polarization *P* depends on the applied external magnetic field *H,* according to the following expression:

$$P = aH \tag{1}$$

where *α* is called second rank ME-susceptibility tensor.

ME effect is quantified by calculating the ME voltage coefficient *α<sup>E</sup>* using the following expression:

$$a\_E = \frac{a}{\mathbf{e}\_o \mathbf{e}\_r} = \left(\frac{\partial \mathbf{E}}{\partial H}\right)\_S \tag{2}$$

Here, *ε<sup>o</sup>* and *ε<sup>r</sup>* are the electric permittivity of free space and the relative permittivity of the dielectric material, respectively. The ME effect was first realized in singlephase materials, the so-called multiferroics, possessing both ferromagnetic and ferroelectric orderings. ME effect is a consequence of the coupling between these ordered parameters. Landau's theory describes this coupling by the free energy *F* of the system expressed in terms of applied field, that is, magnetic field *H* or electric field *E* [21]. The ME effect was theoretically predicted by Curie in 1894 [22]. However, it was first observed by the Russian scientist Astrov in 1960 in a single-phase material Cr2O3, which is antiferromagnetic [23]. Afterward, many more single-phase materials were found to show this effect. ME effect in these single-phase materials was found to be weak and rare due to the nonsimultaneous presence of two ferroic orders at the same temperature. Therefore, artificially engineered multiferroic magnetoelectric composites came into realization. The first bulk ME composite was reported by Van den Boomgaard in the 1970s. Ferroelectric BaTiO3 combined with cobalt ferrite/nickel ferrite to form bulk ME composite produced a ME voltage coefficient, *α<sup>E</sup>* of

130 mV/cm Oe. The constituent materials were synthesized using a solid-state reaction technique [24]. In recent years, different forms of ME composites, such as layered stacks of piezoelectric/magnetostrictive materials, polymer-ceramic matrix composites, and rare earth elements-based composites, which have shown relatively stronger intrinsic ME effect have found great research interest for applications in futuristic electronic devices.

ME effect in the composites of ferromagnetic and piezoelectric materials appears as a product tensor property, which was first proposed by van Suchtelen in 1972. None of the constituent materials used to form a composite shows the ME effect individually. However, on forming a composite, a relatively stronger ME effect results. In this composite, the mechanical deformation in ferromagnetic material due to magnetostriction results in an electrical polarization due to the piezoelectric effect in the piezoelectric material. The product of the magnetostrictive effect (magnetic/mechanical effect) in ferromagnetic material and the piezoelectric effect (mechanical/electrical effect) in piezoelectric material gives a direct ME effect [25]:

Direct ME effect ¼ magnetic*=*mechanical� mechanical*=*electric (3)

$$\text{Converse ME effect} = \text{electric/mechanical}\_{\times} \text{ mechanical/magnetic} \tag{4}$$

A schematic for the ME effect in composites is shown in **Figure 1**. Direct ME effect is observed when the ME composite is subjected to a magnetic field, which causes a change in the shape of the ferromagnetic phase due to magnetostriction. In other words, a strain is developed in ferromagnetic material due to the application of a magnetic field. This strain is then transferred *via* mechanical coupling to the adjacent piezoelectric phase, causing an electric polarization in it due to the piezoelectric effect.

For the ferromagnetic phase, due to the application of a magnetic field to the ME composite,

$$\left(\frac{\partial \mathbf{S}}{\partial \mathbf{H}}\right)\_T = \mathbf{q} \tag{5}$$

and for the piezoelectric phase

$$\left(\frac{\partial \mathcal{P}}{\partial \mathbf{S}}\right)\_E = d \tag{6}$$

where *S* is the strain,*T* is stress, and *q* (where *q* = *dλ/dH*, *λ* is magnetostriction) and *d* are pseudo piezomagnetic and piezoelectric coefficients, respectively. Pseudo piezomagnetic coefficient is defined as the slope of magnetostriction versus magnetic field curve. For an ME composite, electric polarization due to the application of a magnetic field is found as:

$$\left(\frac{\partial P}{\partial H}\right)\_S = k\_c q d = a \tag{7}$$

where *kc* is a mechanical coupling factor (**0** j j *kc* **1**) for the two phases and *α* is the ME susceptibility of the composite, from which we can calculate the ME voltage coefficient as:

$$\mathbf{a}\_{\rm E} = \frac{\mathbf{a}}{\mathbf{e}\_{\sigma}\mathbf{e}\_{r}} = \left(\frac{\partial \mathbf{E}}{\partial \mathbf{H}}\right)\_{\rm S} \tag{8}$$

#### **Figure 1.**

*Schematic diagram of strain-mediated ME effect in an ME composite.*

where *ε<sup>o</sup>* and *ε<sup>r</sup>* are the electrical permittivity of free space and the dielectric constant of the piezoelectric phase, respectively.

Therefore, an entirely new property comes up in a composite of ferromagnetic and piezoelectric materials since neither of the constituent material shows a magnetoelectric effect. This product property-based ME response has resulted in due to the elastic coupling between the two constituent materials. This ME response of the composite, however, strongly depends on individual material characteristics such as the high pseudo piezomagnetic coefficient of ferromagnetic material and large piezoelectric coefficients of piezoelectric material along with strong mechanical coupling (large *kc*) between the two materials. It can be said that the ME effect is an extrinsic property of these ME composites, which is the function of various extrinsic parameters such as the microstructure of the composite and how two phases couple magnetoelectrically at a ferromagnetic-piezoelectric interface. Moreover, this ME response in these composites is observed in ambient conditions, which makes these composites technically viable. Also, the ME response in these composites is several orders of magnitude larger when compared with single-phase magnetoelectric multiferroics. Various composites of ferromagnetic and piezoelectric materials have been investigated in recent years. These ME composites can become useful for practical applications by utilizing this extrinsic ME effect.

## **3. Candidate materials for ME composites**

For an ME composite, the major requirements from a materials perspective are a high piezoelectric coefficient for the piezoelectric phase and large magnetostriction in the ferromagnetic phase. **Figure 2** shows piezoelectric coefficients and magnetostriction some of the constituent phases investigated for ME composites.

Many more candidate materials have been used to form ME composites. A ferromagnetic material can be chosen from a list of metallic FM materials, such Ni, Fe, or Co, from FM alloys, such as Terfenol-D, Metglas, and Permendure, or FM ceramics, such as Magnetic spinel ferrites, Garnets, and Manganites. All these materials have shown a good ME effect when combined with a piezoelectric material. A piezoelectric material is either an oxide ceramic, such as BaTiO3, PZT, KNN, and PMN-PT, or a polymer such as polyvinylidene fluoride (PVDF).

### **3.1 Phase connectivity schemes**

As mentioned earlier, strong ME response in ME composites above room temperature makes them technologically viable in various applications. Therefore, ME composites have been investigated using various combinations of materials and phase connectivity schemes, which include (1) bulk ceramic ME composites formed using piezoelectric ceramics and ferrites, (2) Bi-phase ME composites by combining FM alloys and piezoelectric materials, (3) three-phase (two FM phases and one piezoelectric or otherwise) ME composites, and (4) nanostructured thin films of ferroelectric and magnetic phases-based ME composite [26]. Newnham et al. introduced the concept of phase connectivity schemes (**Figure 3**) in which we use the notations 0–3, 2–2, and 1–3 for describing the structure of two phases of a bulk composite [27]. In these notations, each number represents the connectivity of the respective phase. For instance, 0–3 notation is used for a particulate composite in which particles of the FM phase (represented by 0) are engrained in the matrix of the piezoelectric phase (represented by 3). In 2–2 type composites, which we also call laminate composites, alternate layers (in form of thin sheets/films) of one phase are bonded with layers of another phase. In a 1–3 type composite, fibers/rods/wires/tubes of the magnetostrictive phase are embedded in the matrix of the piezoelectric phase. Each of these structures has its advantages and disadvantages.

Among these connectivity schemes, the most useful is the 2–2 type laminate composite due to its ease of fabrication and relatively better ME response. As most of

**Figure 2.**

*Piezoelectric coefficients and magnetostriction of some of the constituent phases.*

#### **Figure 3.**

*Phase connectivity schemes of (a) 0–3 particulate bulk composite, (b) 1–3 fiber-matrix bulk composite, and (c) 2–2 laminate bulk composite.*

the FM materials possess higher conductivity compared to piezoelectric materials (which are usually good dielectrics) in 0–3 type particulate composite, the inhomogeneous distribution of FM particles in piezoelectric matrix leads to more dielectric losses, which eventually reduces ME response. The issue of higher dielectric losses may be overcome in 1–3 type composites; however, the difficulty of fabrication makes these composites less viable. Moreover, their ME response is also relatively low compared to 2–2 type laminate composite. The connectivity schemes can also be employed to make ME composite thin films on suitable substrates. However, the ME response in these ME composite thin films is not as good as it is in their bulk counterpart. This may be attributed to the relatively lower piezoelectric effect in piezoelectric thin films.

## **4. Applications of ME composites**

ME composites find numerous applications from energy harvesting to AC/DC magnetic field sensors, resonators, phase shifters, ME transformers, etc. Here, we will focus on the energy-harvesting aspect of these ME composites. As mentioned earlier, in an ME composite an electric field can be generated by the application of a magnetic field across the composite, which is termed the direct magnetoelectric effect. Now to produce this direct ME effect, both AC and DC magnetic fields need to be applied across the composite. DC magnetic field is the bias field that produces strain in the magnetostrictive phase through magnetostriction, while the AC field is required to lift the time-reversal symmetry of the magnetostrictive phase. The sweeping of the DC magnetic field initially increases the magnetostriction, which finally reaches saturation at a specific value of the magnetic field for a particular FM material. To generate continuous electrical output from piezoelectric material, we need to subject it to periodic stress. Therefore, in an ME composite, the application of an AC magnetic field leads to the production of periodic stresses in the piezoelectric phase, which is connected to the FM phase *via* strain-mediated magnetoelectric coupling. Thus, an ME composite needs to be subjected to a low-frequency AC magnetic field of small magnitude to generate continuous ME output. This low-frequency weak AC magnetic field is omnipresent in the proximity of power cables. A power cable carrying 50 A alternating current of the frequency 50–60 Hz will generate an alternating magnetic field of about 10 Oe at a distance of 1 cm [28]. This is a weak magnetic field that is usually considered noise and it is also detrimental to the human body. This unused and wasted magnetic field energy could be converted into electrical energy, which will be useful for powering low-power devices.

## **4.1 ME composites-based cantilever**

There have been several designs proposed for effectively harvesting the electrical energy from the magnetic field and acoustic vibrations. The most useful design is to have an ME composite cantilever in which an FM layer is bonded with a piezoelectric layer as a 2–2 type laminate composite. In the case of a cantilever made from piezoelectric material only and subjected to vibrations, a tip mass is used to actuate it. In an ME composite cantilever, the tip mass is replaced with a permanent magnet. On application of an alternating magnetic field across this ME cantilever, the magnetic energy is transformed into vibrations and then into electricity through the piezoelectric effect. A schematic of such an ME cantilever along with a mechanism of the ME effect is shown in **Figure 4**.

Using the above design, Liu et al. [29] demonstrated a copper-based piezoelectric twin beam with a NdFeB permanent magnet of dimensions 30 <sup>6</sup> 0.3 mm<sup>3</sup> . A maximum output power density of 11.73 μW cm<sup>3</sup> Oe<sup>1</sup> was attained with a 100 Hz alternating magnetic field. A similar design was used by Lu et al. [30] for a composite beam of piezoelectric single crystal and Ni. They achieved a power density of 270 μW cm<sup>3</sup> Oe<sup>1</sup> in a 50 Hz AC magnetic field. This kind of design has attracted a lot of attention for energy harvesting applications due to its flexibility and ease of fabrication [31, 32].

## **4.2 Resonant condition ME composite cantilever**

Further enhancement of conversion efficiency can be achieved using ME composites at resonant conditions where the frequency of the AC magnetic field is matched with the natural frequency of the ME composite cantilever. This is usually achieved by making light and slender cantilevers formed from a good mechanical property magnetostrictive material and a strong piezoelectric material. This offers the advantage that it is capable of not only converting vibration energy but also it can harness the magnetic field energy at a low frequency using the ME effect. A schematic of such a dual-phase energy harvester is shown in **Figure 5**. Zhou et al. [30] demonstrated a dual-phase energy harvester of Ni and piezoelectric macro-fiber composites, which produced a power density of 4.5 mW cm<sup>3</sup> G<sup>1</sup> at resonance.

**Figure 5.** *A self-biased dual-phase ME energy harvester.*

## **4.3 Magneto-mechano-electric (MME)-based ME composite energy harvester**

As mentioned earlier, if the tip mass in an ME composite cantilever is replaced with a magnet, it is called a magneto-mechano-electric (MME) component. A schematic of an MME is shown in **Figure 6**.

In this MME component, electrical energy generation is ascribed to three mechanisms: magnetostriction in the FM layer due to the magnets and the deformation of the piezoelectric layer due to magnetostrictive strain, and AC magnetic field-induced vibrations. An MME component of nickel and piezoelectric single-crystal fibers which could generate a power density of 46 mW cm<sup>3</sup> Oe<sup>1</sup> in a magnetic field of 1.6 G was reported by Ryu et al. [33]. This MME component could ignite 35 LEDs. The output power density of ME composites mainly depends on constituent piezoelectric and magnetostrictive materials and also the extent of ME coupling between them. For example, Terfenol-D, which is known as the giant magnetostrictive material and has saturated magnetostriction of around 1500–2000 ppm under the driving magnetic

**Figure 6.** *Schematic of an MME generator.*

*Magnetoelectric Composites-Based Energy Harvesters DOI: http://dx.doi.org/10.5772/intechopen.110875*

field, was used in early ME composite studies [34]. However, its brittleness makes it inviable for low-frequency magnetic field environments due to its higher natural frequency. Moreover, Terfenol-D requires a DC magnetic field of 2–3 kOe to reach saturation magnetostriction [35]. A strong DC magnetic field often leads to bulky structures and usually causes electromagnetic interference as well. Therefore, ME composites requiring a small bias field caught attention [36]. Chu et al. [37] reported Metglas and piezoelectric single-crystal fibers ME composites, which were longitudinally magnetized and poled (L-T) in the transverse direction. A large ME coefficient of 22.92 V cm<sup>1</sup> Oe<sup>1</sup> for a 6 Oe DC bias field could be obtained. However, achieving resonance at low frequency is rather difficult. Therefore, it may be concluded that all these combinations come with certain advantages and disadvantages [38, 39].

## **5. Conclusions**

The ME composites of a ferromagnetic material and a piezoelectric material use the magnetoelectric effect in conjunction with the piezoelectric effect in electrical energy harvesting. They have shown better conversion efficiency as compared to individual piezoelectric energy harvesters. However, the choice of constituent materials and their connectivity schemes play a crucial role in designing these ME composite energy harvesters. Further, enhancement in energy harvesting properties in these ME composites may be achieved by using different phase connectivity schemes. Although the 1–3 phase connectivity scheme has shown significant improvement in the ME effect, the difficulty in the fabrication of these 1–3 type composites is still a hindrance.

## **Acknowledgements**

The authors would like to thank VIT management for their encouragement and support.

## **Conflict of interest**

Authors have no mutual conflict of interest.

## **Appendices and nomenclature**


## **Author details**

Tarun Garg\* and Lickmichand M. Goyal Department of Physics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India

\*Address all correspondence to: gargphy1981@gmail.com

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Magnetoelectric Composites-Based Energy Harvesters DOI: http://dx.doi.org/10.5772/intechopen.110875*

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## **Chapter 5**

## Equivalent Circuit Model of Magnetostrictive/Piezoelectric Laminated Composite

*María Sol Ruiz and Adrián César Razzitte*

## **Abstract**

An equivalent circuital model of magnetostrictive/piezoelectric laminated composite has been developed in order to predict its behavior in presence of dynamic electromagnetic fields. From magnetostrictive and piezoelectric constitutive equations, and using an equation of motion, magnetic-mechanical-electric equations are: obtained by building a symmetric adhoc equivalent circuit about the magnetoelectric (ME) coupling. The coefficients of the direct and converse effects are simulated. The circuit is further used to predict the voltage coefficients of laminated composite. The multilayer material is found to have significantly higher ME coefficients near resonance frequency. The ME coefficients and the voltage coefficients change significantly with the configuration of the multilayer, more specifically when the laminate operates in longitudinal-transverse (L-T) and transverse-transverse (T–T) modes.

**Keywords:** magnetoelectric effect, magnetoelectric voltage coefficient, magnetostrictive, piezoelectric, laminated composite

## **1. Introduction**

The Magnetoelectric (ME) effect arises in multiferroic materials that are electrically and magnetically polarizable due to coupling between electrical polarization and magnetization. These materials are interesting for technological applications, such as magnetic field sensors, transducers, resonators, and devices that provide opportunities in the area of renewable and sustainable energy through energy harvesting [1, 2].

Although the coupling can have nonlinear components, the ME effect is usually described mathematically by the linear ME coupling coefficient, which is the dominant coupling term [3]. The ME effect is characterized by an electric polarization induced by an applied magnetic field or a magnetization induced by an applied electric field called direct magnetoelectric effect (DME) and converse magnetoelectric effect (CME), respectively. The coefficients of the direct (*αDME*) and converse (*αCME*) effects can be expressed as:

$$a\_{\rm DME} = \frac{\partial P}{\partial H} \tag{1}$$

$$a\_{\rm CME} = \frac{\partial B}{\partial E} \tag{2}$$

being *E* electric field intensity, *B* magnetic flux density, *P* electric polarization, and *H* magnetic field intensity.

Maxwell's equations of classical electromagnetism state that, in free space, *E* and *B* are not independent but are intimately linked to each other<sup>1</sup> . Similarly, a theoretical method for the solid state that simultaneously describes relationships *P* � *H* and *B* � *E* (i.e. it describes the DME and CME behaviors) is developed below.

Multilayered composites can have even higher magnetoelectric coefficients than the particulate composites [4] and are therefore considered as viable alternatives [5].

## **2. Modes of operation and vibrational modes**

The system to be modeled consists of bonded ferrite-ferroelectric layers of rectangular geometry. The dimensions are considered to be such that the width *w* and thickness *t* are very small relative to the length *ℓ*; that is, the system is considered as one-dimensional (1D). The thicknesses of the magnetostrictive and piezoelectric layers are designated as *tm* and *tp*, respectively. It is assumed that the interface (boundary between layers) is continuous.

In the modeling of ME laminate, different modes of operation or configurations need to be considered; in addition, to properly defining the coordinate systems (local and global) to be used. For historical reasons, the modes of operation find different lexicons. To prevent confusion, the notation adopted in this work is detailed below. Four basic modes of operation are distinguished and defined, according to the directions of sensed<sup>2</sup> magnetic field (*H*) and electric field (*E*). When the magnetic and electric field directions coincide with the longitudinal direction of the laminate, the mode is defined as "L-L". Similarly, the modes "L-T," "T-L," and "T–T" designate longitudinal magnetic field - transverse electric field, transverse magnetic field longitudinal electric field, and transverse magnetic field - transverse electric field, respectively. The basic modes of operation are outlined in a summary in **Figure 1**.

When a magnetic field is applied to an ME laminate, more than one magnetostrictive vibrational mode is excited. However, the problem can be simplified, without losing correlation with reality, by considering a material with dimensions such that its width and thickness are very small compared to its length. In such case, the longitudinal axis direction of the laminate can be considered as the reference direction. Thus, in this chapter, the main direction of vibration is the longitudinal direction of the laminate<sup>3</sup> (this justifies the selection of a one-dimensional 1D model).

Once the direction of vibration of the laminate is designated, it is then possible to carry out a consistent analysis of the system. When the material is excited with a

<sup>1</sup> The use of the existing connection between electricity and magnetism is what has generated the enormous impact known in the field of technology.

<sup>2</sup> It refers to "sensed" *H* since the lexicon is adopted from the field of application of magnetic sensors; in this notation, the first letter refers to the direction of the magnetic field and the second letter refers to the direction of the electric field.

<sup>3</sup> It is chosen to assign the main vibration direction of the laminate as the reference direction, following the definitions for piezoelectric vibration modes [6].

*Equivalent Circuit Model of Magnetostrictive/Piezoelectric Laminated Composite DOI: http://dx.doi.org/10.5772/intechopen.107749*

**Figure 1.** *Modes of operation of laminated magnetoelectric material.*

magnetic field perpendicular to the longitudinal direction of the laminate, the fundamental mode of vibration will be the transverse mode and when it is excited with a magnetic field parallel to the longitudinal direction of the laminate, the fundamental mode of vibration will be the longitudinal mode (in both cases, as was mentioned, other modes of vibration can be considered as negligible).

## **3. Constitutive equations of the piezoelectric and magnetostrictive phases**

If the system is considered in T–T mode of operation, the direction of polarization in the piezoelectric layer and the direction of magnetic field are along the thickness of the laminate (direction that is made to coincide with the cartesian coordinate axis z). The displacement *u* ¼ *u x*ð Þ coincides with the longitudinal direction of the laminate (direction of the coordinate axis x). **Figure 2a** shows the directions of vectors *D* (electric displacement or electric flux density), *B* (magnetic flux density), *S* (mechanical strain), *T* (mechanical stress), along with the global coordinates x,y,z.

The ME effect of composite materials is due to the combined action of multiple intrinsic characteristics of the materials involved. In materials composed of two phases, two sets of constitutive equations are: required to describe the ME product property. Constitutive equations, in their most general form of the tensor type, are reduced to linear equations (both piezoelectric and piezomagnetic). The equations are: established from an electro/magneto-static point of view (i.e.,, for small excitation signals), and the simplifications are mainly a result of the crystal symmetry of the materials [7]. In the treatment of piezoelectric and piezomagnetic constitutive equations, the subscripts were established taking into account the reference axes used to define piezoelectric coefficients. To comply with this definition, the local coordinates

**Figure 2.**

*(a) Schematic illustration of the laminated bilayer ME in T–T mode and local coordinates for the (b) piezoelectric and (c) piezomagnetic layer.*

shown in **Figure 2b** are defined in the piezoelectric layer and the local coordinates shown in **Figure 2c** are defined in the piezomagnetic layer (i.e., for the magnetic material has established a direction of magnetization<sup>4</sup> in the direction of the axis 3).

### **3.1 Constitutive equations of the piezoelectric laminate**

The constitutive equations of the piezoelectric material for the strain *S*1*<sup>p</sup>* and the electric displacement *D*3, are:

$$S\_{1p} = \mathfrak{s}\_{11}^{E} T\_{1p} + d\_{31,p} E\_3 \tag{3}$$

$$D\_3 = d\_{31,p} T\_{1p} + e\_{33}^T E\_3 \tag{4}$$

Being *E*<sup>3</sup> the electric field in the piezoelectric layer along the z direction, *T*1*<sup>p</sup>* the stress in the piezoelectric layer along the x direction, *s E* <sup>11</sup> the elastic compliance at constant *E*, *d*31,*<sup>p</sup>* the piezoelectric constant, and *ε<sup>T</sup>* <sup>33</sup> the dielectric permittivity at constant stress. The subscript *p* refers to the piezoelectric phase.

### **3.2 Constitutive equations of the magnetostrictive laminate**

The constitutive equations of the magnetostrictive material for the strain *S*1*<sup>m</sup>* and the magnetic flux density *B*3, are:

$$S\_{1m} = s\_{11}^H T\_{1m} + d\_{31,m} H\_3 \tag{5}$$

$$B\_3 = d\_{31,m} T\_{1m} + \mu\_{33}^T H\_3 \tag{6}$$

Being *H*<sup>3</sup> the magnetic field intensity in the magnetostrictive layer along the zdirection, *T*1*<sup>m</sup>* the stress in the piezomagnetic layer along the x direction, *s H* <sup>11</sup> the elastic compliance to constant *H*, *d*31,*<sup>m</sup>* the piezomagnetic constant, and *μ<sup>T</sup>* <sup>33</sup> the magnetic permeability at constant stress. The subscript *m* refers to the magnetostrictive phase.

There is a widely used notation in the design of laminated sensors in which the fundamental vibration mode is designated by two numbers: the first number indicates the direction of the excitation field and the second number indicates the direction of the longitudinal axis (both defined through local coordinates). Taking into account the **Figure 2**, the fundamental vibration mode of the magnetostrictive phase can be called "31-mode" and the fundamental vibration mode of the piezoelectric phase can be called "31-mode". That is, the phases of the composite material will be modeled as magnetostrictive and piezoelectric transducers, both in mode 31. Note the correspondence that exists (for each phase) between its fundamental mode of vibration and the subscripts of its "piezo-" parameters in the constitutive equations<sup>5</sup> .

<sup>4</sup> Although strictly speaking, magnetization in ferrites is a magnetic polarization phenomenon, and polarization in piezoelectrics is an electrical polarization phenomenon, here "magnetization" is used for the magnetostrictive layer and "polarization" for the piezoelectric layer.

<sup>5</sup> The term "piezo-" refers to the piezoelectric or piezomagnetic coefficients, which are precisely electromechanical and magneto-mechanical parameters, respectively.

## **4. Equation of motion: general solution and boundary conditions**

In order to design laminates operating under dynamic conditions, the coupling between the composite layers is considered through an equation of motion. Considering that the applied external field (electric or magnetic) is sinusoidal, the vibrational movement of the laminate will also be sinusoidal (harmonic oscillator).

The differential equation of motion for any element of mass can be obtained using Newton's second law, written as:

$$\overline{\rho}\frac{\partial^2 u(\varkappa, t)}{\partial t^2} = r\_m \frac{\partial T\_{1m}}{\partial \varkappa} + r\_p \frac{\partial T\_{1p}}{\partial \varkappa} \tag{7}$$

Where *u x*ð Þ , *t* is the displacement function of the medium and *rm* and *rp* are the relative relations of thickness of the magnetostrictive and piezoelectric layers, respectively, that is: *rm* ¼ *tm=t* and *rp* ¼ *tp=t*, and it is evident that *rm* þ *rp* ¼ 1. Furthermore, *ρ* ¼ *rp ρ<sup>p</sup>* þ *rm ρ<sup>m</sup>* is the average mass density, with *ρ<sup>p</sup>* and *ρ<sup>m</sup>* as the mass densities of the piezoelectric and magnetostrictive components, respectively.

In the 1D model, mechanical movements are considered only in the longitudinal direction of the laminate (x-axis). In the direction perpendicular to it, the component layers are considered as free bodies and for the model, there are no strains or stresses between the layers of the composite.

The magnetostrictive and piezoelectric mass elements share the same displacement component *u* and the same strain component, that is:

$$
\mathfrak{u}\_m = \mathfrak{u}\_p = \mathfrak{u} \tag{8}
$$

$$\mathcal{S}\_{1m} = \mathcal{S}\_{1p} = \mathcal{S} = \frac{\partial u}{\partial \mathbf{x}} \tag{9}$$

From Eq. (3) and (5), the mechanical stresses in the phases can be expressed as:

$$T\_{1p} = \frac{S\_{1p}}{s\_{11}^E} - \frac{d\_{31,p}}{s\_{11}^E} E\_3 \tag{10}$$

$$T\_{1m} = \frac{1}{s\_{11}^H} \mathbf{S}\_{1m} - \frac{d\_{31,m}}{s\_{11}^H} H\_3 \tag{11}$$

Substituting in Eq. (7) the derivatives of Eq. (10) and (11) and considering that ð Þ *<sup>∂</sup>E*<sup>3</sup> *<sup>=</sup>∂<sup>x</sup>* <sup>¼</sup> ð Þ *<sup>∂</sup>H*<sup>3</sup> *<sup>=</sup>∂<sup>x</sup>* <sup>¼</sup> 0 (since both *<sup>E</sup>*<sup>3</sup> and *<sup>H</sup>*<sup>3</sup> do not vary along the length of the material), the equation of motion is written as:

$$\frac{\partial^2 u(\mathbf{x},t)}{\partial t^2} = \overline{\nu}^2 \frac{\partial^2 u(\mathbf{x},t)}{\partial \mathbf{x}^2} \tag{12}$$

where *ν* is the speed of sound in the composite material:

$$
\overline{\nu}^2 = \left(\frac{r\_m}{s\_{11}^H} + \frac{r\_p}{s\_{11}^E}\right) / \overline{\rho} \tag{13}
$$

Eq. (12) is the classical wave equation of motion. Your resolution makes it simplified to:

*Novel Applications of Piezoelectric and Thermoelectric Materials*

$$\frac{\partial^2 u(\mathbf{x})}{\partial \mathbf{x}^2} + k^2 u(\mathbf{x}) = \mathbf{0} \tag{14}$$

Which has the form of the equation of motion of a simple harmonic oscillator but with the spatial variable instead of temporal. In addition, the so-called dispersion relation *k* ¼ *ω=ν* holds, where *k* is the wave number and *ω* is the angular frequency.

The general solution of Eq. (14) is:

$$u(x) = A\cos kx + B\sin kx \tag{15}$$

To solve the integration constants *A* and *B*, the boundary conditions expressed in terms of displacement speeds *u*\_ are established: considering that the ends of the material are free of external stresses, let *u*\_ <sup>1</sup> be the displacement speed at *x* ¼ 0 and *u*\_ <sup>2</sup> the displacement speed in *x* ¼ *ℓ*, then:

$$
\dot{u}\_1 = jau(0) \tag{16}
$$

$$
\dot{u}\_2 = jau(\ell) \tag{17}
$$

The resolution leads to the expression:

$$u(\mathbf{x}) = \frac{\dot{u}\_1}{j\alpha} \cos k\mathbf{x} + \frac{\dot{u}\_2 - \dot{u}\_1 \cos k\ell}{j\alpha \sin k\ell} \sin k\mathbf{x} \tag{18}$$

And, according to Eq. (9) and (18), the strains at the faces of the laminate are:

$$S(\mathbf{x} = \mathbf{0}) = \frac{\dot{u}\_2 - \dot{u}\_1 \cos k\ell}{j\overline{\nu}\sin k\ell} \tag{19}$$

$$S(\varkappa = \ell') = \frac{\dot{u}\_2 \cos k\ell - \dot{u}\_1}{\dot{\mathcal{V}} \sin k\ell} \tag{20}$$

## **5. Potential: current coupling**

Substituting Eq. (10) in Eq. (4), is obtained:

$$D\_3 = d\_{31,p} \left( \frac{\mathcal{S}\_{1p}}{s\_{11}^E} - \frac{d\_{31,p}}{s\_{11}^E} E\_3 \right) + \varepsilon\_{33}^T E\_3 \tag{21}$$

By definition of electric potential and taking into account the geometry of the system, it can be expressed that the piezoelectric vibrator is satisfied: *Vp* ¼ *tp E*<sup>3</sup> . Furthermore, if defined:

$$\overline{\varepsilon\_{33}} = \varepsilon\_{33}^{T} \left( 1 - \frac{d^2 \mathbf{1}\_{31,p}}{\varepsilon\_{11}^{E} \varepsilon\_{33}^{T}} \right) \tag{22}$$

Then the electrical displacement can be written as:

$$D\_3 = \frac{d\_{31,p}}{s\_{11}^E} \ S\_{1p} + \overline{\varepsilon\_{33}} E\_3 \tag{23}$$

*Equivalent Circuit Model of Magnetostrictive/Piezoelectric Laminated Composite DOI: http://dx.doi.org/10.5772/intechopen.107749*

The electric charge at the surface of the piezoelectric is obtained integrating over the surface, according to:

$$Q = \int\_{0}^{\ell} \int\_{0}^{\bullet} D\_{3} \, d\boldsymbol{x} d\boldsymbol{y} = \boldsymbol{\omega} \frac{d\_{31,p}}{s\_{11}^{E}} [\boldsymbol{u}(\ell') - \boldsymbol{u}(\mathbf{0})] + \boldsymbol{\omega} \overline{\epsilon\_{33}} E\_{3} \tag{24}$$

From Eq. (18), the expressions for *u*ð Þ *ℓ* and *u*ð Þ 0 are obtained according to:

$$u(\ell) = \frac{\dot{u}\_1}{j\dot{o}o} \cos k\ell + \frac{\dot{u}\_2 - \dot{u}\_1 \cos k\ell}{j\dot{o}o \sin k\ell} \sin k\ell = \frac{\dot{u}\_2}{j\dot{o}o} \tag{25}$$

$$u(\mathbf{0}) = \frac{\dot{u}\_1}{j\alpha} \cos k\mathbf{0} + \frac{\dot{u}\_2 - \dot{u}\_1 \cos k\ell}{j\alpha \sin k\ell} \sin k\mathbf{0} = \frac{\dot{u}\_1}{j\alpha} \tag{26}$$

Then:

$$Q = \int\_{0}^{\ell} \int\_{0}^{\bullet} D\_{3} \, d\mathbf{x} dy = \omega \frac{d\_{31,p}}{j \alpha \varepsilon\_{11}^{E}} (\dot{u}\_{2} - \dot{u}\_{1}) + \omega \epsilon \overline{\varepsilon\_{33}} E\_{3} \tag{27}$$

Taking into account the properties of harmonic phasor (the analysis is in the field of angular frequency, not temporal), the electric current in the piezoelectric vibrator *Ip* can be expressed as:

$$I\_p = \frac{dQ}{dt} = j\alpha Q \tag{28}$$

Defining the static capacitance in the piezoelectric layer *C*<sup>0</sup> and the electromechanical coupling factor *φ<sup>p</sup>* as:

$$C\_0 = \frac{\overline{\varepsilon\_{33}} \omega}{t\_p} \tag{29}$$

$$
\rho\_p = \omega \frac{d\_{31,p}}{s\_{11}^E} \tag{30}
$$

And substituting Eq. (29) and (30) in Eq. (28), the expression of coupling for piezoelectric material is:

$$I\_p = j\alpha \mathbf{C}\_0 V\_p + \rho\_p (\dot{u}\_2 - \dot{u}\_1) \tag{31}$$

Similarly, from the constitutive equations of the magnetostrictive material, replacing Eq. (11) in (6), is obtained:

$$B\_3 = \left(\frac{d\_{31,m}}{s\_{11}^H}\right) S\_{1m} + \overline{\mu\_{33}^T} H\_3 \tag{32}$$

where is defined:

$$\overline{\mu\_{33}^T} = \mu\_{33}^T \left( 1 - \frac{d^2 \mathbf{1}\_{31,m}}{\mu\_{33}^T \mathbf{s}\_{11}^H} \right) \tag{33}$$

According to Faraday's law of induction, on the solenoid *Vm* can be expressed as:

$$V\_m = -N\frac{d\Phi}{dt}\tag{34}$$

where Φ ¼ *B*3*wtm* is the magnetic flux. And under concepts of harmonic phasors, the differential of the magnetic flux can be expressed as:

$$V\_m = -Nj\alpha\Phi = -Nj\alpha B\_3 \omega t\_m \tag{35}$$

Substituting Eq. (32) in Eq. (35):

$$V\_m = -Nj\omega\omega\_m \left[ \left( \frac{d\_{31,m}}{s\_{11}^H} \right) S\_{1m} + \overline{\mu\_{33}^T} H\_3 \right] \tag{36}$$

Considering a solenoid with *N* turns of total length *ℓ* and through which an electric current *Im* circulates, the generated magnetic field in the solenoid can be expressed as:

$$H\_3 = \frac{N}{\ell} I\_m \tag{37}$$

And taking into account that the relationship between deformation and displacement velocities can be expressed by the relation [8]:

$$S\_{1m} = \frac{\dot{u}\_2 - \dot{u}\_1}{j\alpha\ell} \tag{38}$$

Then:

$$V\_m = -\frac{N\omega t\_m}{\ell} \frac{d\_{31,m}}{s\_{11}^H} (\dot{u}\_2 - \dot{u}\_1) - j\alpha \frac{\omega t\_m}{\ell} \overline{\mu\_{33}^T} I\_m \tag{39}$$

Defining the static inductance in the magnetostrictive layer *L*<sup>0</sup> and the magnetomechanical coupling factor *φ<sup>m</sup>* as:

$$L\_0 = \overline{\mu\_{33}^T} N^2 \frac{\mathfrak{a} t\_m}{\ell} \tag{40}$$

$$
\rho\_m = \omega t\_m \, \frac{N}{t'} \, \frac{d\_{31,m}}{s\_{11}^H} \tag{41}
$$

The expression of coupling for magnetostrictive material is:

$$V\_m = -\rho\_m(\dot{u}\_2 - \dot{u}\_1) - j\alpha L\_0 I\_m \tag{42}$$

## **6. Magneto-mechano-electric coupling**

Based on the constitutive equations of both components (Section 3) and the equation of motion (via strain–stress coupling between layers; Section 4), an equivalent circuit model for laminated ME is developed. The mechanical forces are relative to the stresses. If the forces at the ends of the laminate are called *F*<sup>1</sup> at *x* ¼ 0 and *F*<sup>2</sup> at *x* ¼ *ℓ*, it is satisfied that:

*Equivalent Circuit Model of Magnetostrictive/Piezoelectric Laminated Composite DOI: http://dx.doi.org/10.5772/intechopen.107749*

$$-F\_1 = \left. \mathfrak{a} t\_p \left. T\_{1p} \right|\_{\mathfrak{x}=0} + \left. \mathfrak{a} t\_m \left. T\_{1m} \right|\_{\mathfrak{x}=0} \right. \tag{43}$$

$$-F\_2 = \left. \mathfrak{a} t\_p \left. T \right|\_{\mathfrak{x} = \ell} + \left. \mathfrak{a} t\_m \left. T \right|\_{\mathfrak{x} = \ell} \right| \tag{44}$$

Substituting Eq. (10), (11), (19), and (20) in Eq. (43) and (44) and taking into account Eq. (18), the expressions are: obtained:

$$F\_1 = -\left(\frac{\omega t\_p}{s\_{11}^E} + \frac{\omega t\_m}{s\_{11}^H}\right)\left(\frac{\dot{\imath}\_2 - \dot{\imath}\_1 \cos k\ell}{j\overline{\nu}\sin k\ell}\right) + \omega t\_p \left.\frac{d\_{31,p}}{s\_{11}^E} E\_3 + \omega t\_m \frac{d\_{31,m}}{s\_{11}^H} H\_3 \right. \tag{45}$$

$$F\_2 = -\left(\frac{\omega t\_p}{s\_{11}^E} + \frac{\omega t\_m}{s\_{11}^H}\right)\left(\frac{\dot{u}\_2 \cos k\ell - \dot{u}\_1}{j\overline{\nu}\sin k\ell}\right) + \omega t\_p \frac{d\_{31,p}}{s\_{11}^E} E\_3 + \omega t\_m \frac{d\_{31,m}}{s\_{11}^H} H\_3 \tag{46}$$

Furthermore, with the equations obtained in Section 5, the force expressions can be simplified to:

$$F\_1 = -\overline{\rho}\overline{\nu}\omega t \left(\frac{\dot{u}\_2 - \dot{u}\_1 \cos k\ell}{j \sin k\ell}\right) + \rho\_p V\_p + \rho\_m I\_m \tag{47}$$

$$F\_2 = -\overline{\rho}\overline{\nu}\omega t \left(\frac{\dot{u}\_2\cos k\ell - \dot{u}\_1}{j\sin k\ell}\right) + \rho\_p V\_p + \rho\_m I\_m \tag{48}$$

If mechanical impedances are defined as:

$$Z\_1 = \frac{\overline{\rho}\,\overline{\nu}\,\mathfrak{w}t}{j\sin kt} \tag{49}$$

$$Z\_2 = j\overline{\rho}\,\overline{\nu}\,\omega t \,\tan\frac{k\ell}{2} \tag{50}$$

Then the mechanical forces are rewritten as:

$$F\_1 = Z\_1(\dot{\mu}\_1 - \dot{\mu}\_2) + Z\_2\dot{\mu}\_1 + \rho\_p V\_p + \rho\_m I\_m \tag{51}$$

$$F\_2 = Z\_1(\dot{u}\_1 - \dot{u}\_2) - Z\_2\dot{u}\_2 + \rho\_p V\_p + \rho\_m I\_m \tag{52}$$

The ME equivalent circuit corresponding to the magneto-mechano-electric expressions obtained is represented in **Figure 3**. In this circuit, the forces *F*<sup>1</sup> and *F*<sup>2</sup> act as "mechanical voltage" and the displacements *u*\_ <sup>1</sup> and *u*\_ <sup>2</sup> as "mechanical currents". Thus, an applied magnetic field produces a"mechanical voltage" via the magnetomechanical coupling factor *φ<sup>m</sup>* and due to the electro-mechanical coupling factor *φp*,

$$\begin{array}{c|cc} \underline{\dot{u}\_{1}} & \underline{Z\_{2}} & \underline{Z\_{2}} & \dot{u}\_{2} \\ \hline \hline F\_{1} \downarrow & \underline{I\_{p}} & \underline{\dot{0}} & \underline{F\_{2}} \\ V\_{p} \underline{\dot{\phantom{0.0pt}{\cdot}}} & \underline{C\_{0}} & \underline{\dot{1}} \\ V\_{m} \underline{\dot{\phantom{0.0pt}{\cdot}}} & \underline{\dot{0}} & \underline{1} : \varphi\_{p} \\ \hline \end{array} \bigg| 1; \varphi\_{p} \right| $$

**Figure 3.** *Magneto-mechano-electric equivalent circuit.*

$$V\_m \xleftarrow{I\_m} \xrightarrow{I\_1} \xleftarrow{Z\_1 \underset{L\_0}{\longleftarrow}} \xleftarrow{Z\_2/2} \xleftarrow{I\_p} V\_p$$

**Figure 4.** *Magneto-mechano-electric equivalent circuit under condition of free edges.*

$$\varphi\_m \nu\_m \xrightarrow[]{I\_m/\varphi\_m} \xrightarrow[]{I\_m/\varphi\_m \quad \quad I\_p/\varphi\_p} \xrightarrow[]{\mathbb{L}} \varphi\_p V\_p$$

**Figure 5.**

*Magneto-mechano-electric equivalent circuit under free edge condition simplified.*

generates an electrical potential difference across the piezoelectric layer. In the circuit model of **Figure 4**, a transformer with ratio 1 : *φ<sup>p</sup>* is used to represent the electromechanical coupling. Similarly, an applied electric field due to the magnetomechanical coupling factor *φm*, generates a magnetization in the magnetostrictive layer. In the circuit model of **Figure 4**, a transformer with ratio 1 : *φ<sup>m</sup>* is used to represent the magneto-mechanical coupling. Note that, the equivalent circuit model considers both electro-mechanical coupling and magneto-mechanical coupling.

Under free-free boundary conditions, that is, null forces at *x* ¼ 0 and *x* ¼ *ℓ* (*F*<sup>1</sup> ¼ *F*<sup>2</sup> ¼ 0Þ, the input and output terminals are grounded, and the circuit is simplified to the one shown in **Figure 5** and then simplified to that shown in **Figure 3**.

The mechanical impedance is: *Zmech* ¼ *Z*<sup>1</sup> þ *Z*2*=*2. Therefore, from Eq. (49) and (50), is obtained:

$$Z\_{mech} = -\frac{j}{2}\overline{\rho}\,\overline{\nu}\,\omega t\,\text{cot}\,\frac{k\ell}{2} \tag{53}$$

As the equivalent ME circuit obtained (**Figure 3**) contains symmetrical parts (magnetic and electrical), is possible to discuss both direct and converse effects. To analyze these effects, "open-circuit" and "short-circuit" conditions are: imposed:

Converse ME effect: Under open circuit condition *Ip* ¼ *Im* ¼ 0, the following relationship is derived:

$$\frac{\rho\_m dV\_m}{\rho\_p dV\_p} = \frac{j\alpha L\_0 \rho\_m^2}{j\alpha L\_0 \rho\_m^2 + Z\_1 + Z\_2/2} \tag{54}$$

Taking into account that: *Vm* ¼ �*NjωwtmB*<sup>3</sup> in the solenoid and *Vp* ¼ *tpE*<sup>3</sup> in the piezoelectric vibrator, and according to Eq. (2), the converse ME coefficient is expressed as:

$$\alpha\_{\rm CME(T-T)} = \frac{\partial B}{\partial E} = -\frac{\frac{d\_{31,m}}{s\_{11}^d \mu\_{33}^T} \frac{d\_{31,p}}{s\_{11}^E} r\_p}{\left(\frac{d\_{31,n}}{s\_{11}^d \mu\_{33}^T}\right)^2 r\_m + \frac{\alpha \ell}{2\mu\_{33}^T} \overline{\rho} \,\overline{\nu} \,\mathrm{cot}\,\frac{k\ell}{2}}}\tag{55}$$

By indicating *αCME* with the subscript T–T, it has been possible to omit the subscripts of *B* and *E*.

*Equivalent Circuit Model of Magnetostrictive/Piezoelectric Laminated Composite DOI: http://dx.doi.org/10.5772/intechopen.107749*

Direct ME effect: Under short-circuit condition *Vp* ¼ *Vm* ¼ 0, the following relationship is derived:

$$\frac{dI\_p/\rho\_p}{dI\_m/\rho\_m} = \frac{\frac{1}{\rho\_p^{\prime\prime}/j\alpha C\_0 + Z\_1 + Z\_2/2}}{\frac{1}{j\alpha L\_0 \rho\_m^{\prime}} + \frac{1}{\rho\_p^{\prime\prime}/j\alpha C\_0 + Z\_1 + Z\_2/2}}\tag{56}$$

Taking into account Eq. (28) and (37), and according to Eq. (1), the direct ME coefficient is obtained as:

$$\alpha\_{\text{DME}(T-T)} = \frac{\partial P}{\partial H} = \frac{\frac{d\_{\text{N,m}}}{s\_{\text{11}}^H \overline{\mu}\_{\text{\beta}}^T} \frac{d\_{\text{31},p}}{s\_{\text{11}}^H} r\_m}{\left(\frac{d\_{\text{N,m}}}{s\_{\text{11}}^H \overline{\mu}\_{\text{\beta}}^T}\right)^2 r\_m + \frac{r\_p}{\overline{r}\_{\text{33}} \overline{\mu}\_{\text{\beta}}^T} \left(\frac{d\_{\text{31},p}}{s\_{\text{11}}^H}\right)^2 + \frac{ae\ell}{2\overline{\mu}\_{\text{\beta}}^T} \overline{\nu} \,\overline{\nu} \,\text{cot } \frac{k\ell}{2}}\tag{57}$$

By indicating *αDME* with the subscript T–T, it has been possible to omit the subscripts of *P* and *H*.

Thus, expressions of CME and DME coefficients with physical parameters of the phases that constitute the composite material are obtained. They are mathematical expressions that no longer involve circuit parameters but physical parameters of the ferrite and ferroelectric. Eq. (55) and (57) show the expression of the CME and DME coefficients as a function of frequency, however for frequencies much lower than the resonance frequency of the material (*ω* ≪ *ωr*), it can be approximated that cot *<sup>k</sup><sup>ℓ</sup>=*<sup>2</sup> � 2*=<sup>k</sup><sup>ℓ</sup>* and as a result the CME and DME coefficients become independent of the frequency and the length of the laminate.

## **7. ME laminate under resonance frequency**

When the alternating field of excitation (electric or magnetic) has a frequency near to the resonance frequency of the material, Eq. (55) and (57) give an infinite value of ME coefficients, which is physically impossible. To predict the ME behavior near to its resonance frequency, it is necessary to consider the effect of mechanical dissipation in the energy conversion process. The mechanical impedance in the resonant state can be approximated as a series RLC circuit shown in **Figure 6**.

For such a model, the circuit impedance is obtained simplify Eq. (53) by using the Taylor series expansion [9]:

$$Z\_{mech} = R\_{mech} + j\alpha L\_{mech} + \frac{1}{j\alpha C\_{mech}}\tag{58}$$

where *Rmech*, *Lmech*, and *Cmech* are the effective mechanical resistance, inductance and capacitance, respectively, which have the expressions:

$$\rho\_m V\_m \xleftarrow[\underbrace{I\_m/\varphi\_m \quad R\_{mech} \quad L\_{mech} \quad C\_{mech} \quad I\_p/\varphi\_p}\_{\rightleftharpoons} \varphi\_p V\_p$$

**Figure 6.**

*RLC equivalent circuit under resonance frequency.*

$$R\_{mech} = \frac{\pi Z\_0}{8Q\_{mec}}\tag{59}$$

$$L\_{mech} = \frac{\pi Z\_0}{8\alpha\_r} \tag{60}$$

$$C\_{mech} = \frac{8}{\pi \alpha\_r Z\_0} \tag{61}$$

being *Z*<sup>0</sup> ¼ *ρ νwt*; *ω<sup>r</sup>* ¼ *πν=<sup>ℓ</sup>* the average resonance frequency of the composite material (note that it holds: *ω<sup>r</sup>* <sup>2</sup> <sup>¼</sup> *Lmech*�<sup>1</sup> *Cmech*�<sup>1</sup> ) and *Qmech* the mechanical factor (numerically equal to the inverse of the tangent of mechanical losses). As the mechanical losses in the composite material are the result of the mechanical losses in the magnetic phase (1*=Qm*) and piezoelectric phase (1*=Qp*), it can be expressed:

$$\frac{1}{Q\_{mech}} = \frac{r\_m}{Q\_m} + \frac{r\_p}{Q\_p} \tag{62}$$

Therefore considering mechanical dissipation, the expressions for the CME and DME coefficients are obtained according to:

$$a\_{\rm CME}(T-T) = -\frac{\frac{d\_{31,m}^{\rm d\_{31,p}}}{\frac{d\_{31,m}^{\rm d\_{31,p}}}{d\_{31}^{\rm d\_{31}}}} \frac{d\_{31,p}}{r\_m} r\_p}{r\_m + \frac{\pi \overline{\rho} \cdot \overline{\nu} \ell^{\prime} a\_r}{8 \mu\_{33}^{\rm T}} \left(1 - \frac{a^2}{a\_r^{\rm d}}\right) + \frac{j \pi \overline{\rho} \cdot \overline{\nu} a \ell}{8 \mu\_{33}^{\rm T} Q\_{\rm mc}}}\tag{63}$$

$$\alpha\_{\text{DME}}(T-T) = \frac{\frac{d\_{\text{31,m}}^{\text{M,m}}\frac{d\_{\text{31},p}}{d\_{\text{33}}^{\text{M}}}r\_{\text{11}}}{\left(\frac{d\_{\text{31,m}}}{d\_{\text{11}}^{\text{M}}\mu\_{\text{33}}^{\text{T}}}\right)^{2}r\_{m} + \frac{r\_{p}}{\overline{r\_{\text{33}}}\mu\_{\text{33}}^{\text{T}}}\left(\frac{d\_{\text{31},p}}{d\_{\text{11}}^{\text{E}}}\right)^{2} + \frac{\overline{\pi p}\ \overline{\nu}\epsilon a\_{\text{V}}}{8\mu\_{\text{33}}^{\text{T}}}\left(\mathbbm{1} - \frac{\alpha^{2}}{\alpha\_{r}^{2}}\right) + \frac{j\pi\overline{\rho}\ \overline{\nu}\alpha\ell}{8\mu\_{\text{33}}^{\text{T}}Q\_{\text{mcc}}}\tag{64}$$

From Eq. (63) and (64), it can be seen that the CME and DME coefficients have complex expressions relative to frequency.

## **8. Simulation of the ME laminate in T: T mode**

The behavior of a composite material with lead-free barium titanate (BTO) as ferroelectric component and nickel ferrite (NFO) as ferrimagnetic component is simulated. **Table 1** lists the values used in the model for polycrystalline and polarized phases [10–13]. Arbitrarily, a material geometry of length *ℓ* ¼ 20 mm has been selected. With Eq. (63) and (64), the resonance profiles of ∣*αCME*∣ and ∣*αDME*∣ were


**Table 1.**

*Physical parameters of barium titanate (BTO) and nickel ferrite (NFO).*

*Equivalent Circuit Model of Magnetostrictive/Piezoelectric Laminated Composite DOI: http://dx.doi.org/10.5772/intechopen.107749*

obtained as a function of frequency. The programming codes were developed in the Matlab® software.

**Figures 7** and **8** show the results of modules of *αCME* and *αDME* at frequencies from 20 kHz to 90 kHz, with *rm* as a parameter. From these profiles, it is observed that the ∣*αCME*∣ and ∣*αDME*∣ reach their maximum value when they operate near the resonance frequency. In addition, these results show that the ME coupling is strongly dependent on the thickness ratio between piezomagnetic and piezoelectric components, which is poorly reported with experimental data due to the difficulty in preparing the material.

**Figure 9** shows the maxima of ∣*αCME*∣ and ∣*αDME*∣ as a function of the magnetostrictive phase content in the composite material. The cases for *rm* ¼ 0 (without magnetostrictive layer) and *rm* ¼ 1 (without piezoelectric layer) have been omitted since under these conditions there is no magneto-elasto-electric coupling and *αCME* ¼ *αDME* ¼ 0. It is observed that, in absolute values, the DME coefficient

**Figure 7.** ∣*αCME*∣ *at frequencies from 20 kHz to 90 kHz, for ME laminate in T–T mode.*

**Figure 8.** ∣*αDME*∣ *at frequencies from 20 kHz to 90 kHz, for ME laminate in T–T mode.*

#### **Figure 9.**

*Maximum of* ∣*αCME*∣ *and* ∣*αDME*∣ *versus rm for ME laminate in T–T mode.*

increases with the content of the piezomagnetic component, while the CME coefficient decreases.

In addition, it is observed (**Figure 9**) that ∣*αCME*∣ and ∣*αDME*∣ become similar for composite materials with *rm* � 0*:*5. This means that for the cases in which the structures have a similar phase content, the energy transition capacity in both effects is similar (the mechanical stress action is mutual).

The resonance frequency of the CME effect can be obtained when the module of *αCME* reaches its maximum value, so deriving Eq. (63) with respect to the angular frequency ω, the expression is obtained:

$$\rho\_{\rm resCME} = \rho\_r \sqrt{1 + \left(\frac{d\_{31,m}}{s\_{11}^H \mu\_{33}^T}\right)^2 \frac{8\mu\_{33}^{\bar{T}} r\_m}{\pi \overline{\rho} \, \overline{\nu} \ell^e \alpha\_r} - \frac{1}{2Q\_{mcc}^{\circ 2}}} \tag{65}$$

Similarly, the resonance frequency of the DME effect can be obtained when the module of *αDME* reaches its maximum value, so deriving Eq. (64) with respect to the angular frequency ω, the expression is obtained:

$$\rho\_{\rm reDME} = \alpha\_r \sqrt{1 + \left(\frac{d\_{31,m}}{s\_{11}^H \mu\_{33}^T}\right)^2 \frac{8\bar{\mu\_{33}^T} r\_m}{\pi \overline{\rho} \, \overline{\nu} \ell' \alpha\_r} + \left(\frac{d\_{31,p}}{s\_{11}^E}\right)^2 \frac{8 \, r\_p}{\overline{\varepsilon\_{33}} \pi \overline{\rho} \, \overline{\nu} \ell' \alpha\_r} - \frac{1}{2Q\_{\rm mac}}\tag{66}$$

It can be noted that both the CME and DME resonance frequencies are independent of the total thickness of the compound *t*, they only depend on the thickness ratio between the phases.

**Figure 10** shows the resonance frequencies (calculated using the relationship *ω* ¼ 2*πf* ). It is observed that the resonance frequencies (both CME and DME) increase with the ferrite content. In other words, the maxima in the profiles of **Figures 7** and **8** change at higher frequencies with the magnetostrictive phase content. According to Eq. (65) and (66), the value of *ωresDME* is always greater than *ωresCME* due to the term *d*31,*p=s E* <sup>11</sup> � �<sup>2</sup> 8 *rp=ε*33*πρ νℓω<sup>r</sup>* � �. This provides a mathematical interpretation for the profiles obtained in **Figure 10**, where it is observed that the DME resonance frequencies are always greater than the CME resonance frequencies, for any value of *rm*. As seen in

*Equivalent Circuit Model of Magnetostrictive/Piezoelectric Laminated Composite DOI: http://dx.doi.org/10.5772/intechopen.107749*

#### **Figure 10.**

*CME and DME resonance frequencies (f resCME and fresDME) as a function of magnetostrictive phase content, for ME laminate in T–T mode.*

**Figure 10**, the difference between CME and DME resonance frequencies become smaller with the magnetostrictive phase content. That is, *ωresDME* � *ωresCME* for high values of *rm*. This allows to infer that the difference in resonance frequencies is mainly caused by the piezoelectric phase.

Another important parameter to characterize the ME effect is the so-called voltage coefficient, defined as:

$$a\_V = \frac{\partial V}{\partial H} \tag{67}$$

In the piezoelectric phase is satisfied: *P*<sup>3</sup> ¼ *ε*<sup>0</sup> *χ<sup>e</sup> E*3, being *ε*<sup>0</sup> the dielectric permittivity of free space and *χ<sup>e</sup>* the dielectric susceptibility of the material (defined as *χ<sup>e</sup>* ¼ *<sup>ε</sup><sup>T</sup>* 33*=<sup>ε</sup>*<sup>0</sup> � 1). Considering that *Vp* <sup>¼</sup> *tpE*<sup>3</sup> (Section 5) and that *<sup>ε</sup><sup>T</sup>* <sup>33</sup> � *ε*<sup>0</sup> � *<sup>ε</sup><sup>T</sup>* <sup>33</sup> then:

$$a\_V = \frac{\partial V\_3}{\partial H\_3} = \frac{t\_p}{\varepsilon\_0 \chi\_e} \frac{\partial P\_3}{\partial H\_3} = \frac{r\_p t}{\varepsilon\_{33}^T} \frac{\partial P\_3}{\partial H\_3} \tag{68}$$

According to the definition of the DME coefficient (Eq. (1)), the voltage coefficient can be written as:

$$a\_V = \frac{r\_p t}{\varepsilon\_{33}^T} \alpha\_{DME} \tag{69}$$

Note that, unlike *αDME*, the voltage coefficient depends on the thickness *t* of the material. **Figure 11** shows the results obtained from the voltage coefficients at the resonance frequency (maximum voltage coefficients) as a function of the piezomagnetic phase content (considering a total thickness of *t* ¼ 2 mm in Eq. (69)). To express *α<sup>V</sup>* in units of V/Oe (cgs system), the values in SI units were multiplied by the 10<sup>3</sup> *=*4*π <sup>V</sup>=Oe kg m*3*=C*<sup>2</sup> *s* . It is observed that there is an optimum value of magnetic phase content that corresponds to a maximum value of *αV*. More specifically, a maximum of *α<sup>V</sup>* � 80 *mV=Oe* is observed for *rm* ¼ 0*:*3 and at a frequency *f* � 116 *kHz*.

**Figure 11.** *α<sup>V</sup> versus rm, at resonance frequency, for ME composite in T–T mode.*

It is interesting to note that the trend of *α<sup>V</sup>* and *αDME* with the values of *rm* do not coincide with each other. The maximum of *αDME* is obtained when *rm* ¼ 0*:*9 (**Figure 9**), while the maximum in *α<sup>V</sup>* is observed with *rm* ¼ 0*:*3 (**Figure 11**). In the field of application, such as transducer devices, most of the time the physical quantity that is measured is the electric potential, so the voltage coefficient could be considered a more significant parameter.

## **9. Simulation of the ME laminate in L-T mode**

If the analysis is extended to an L-T configuration (**Figure 1**), the model must now imply a structure, where the polarization direction of the piezoelectric phase is transverse to the longitudinal direction of the material and the applied magnetic field is longitudinal. **Figure 12a** shows the directions of vectors *D* (electric displacement), *B* (magnetic flux density), *S* (mechanical strain), *T* (mechanical stress), along with the global coordinates x,y,z. The local coordinates shown in **Figure 12b** are defined in the piezoelectric layer and the local coordinates shown in **Figure 12c** are defined in the piezomagnetic layer.

In L-T mode, the constitutive equations of the piezoelectric material are Eq. (3) and (4) and the constitutive equations of the piezomagnetic material for the strain *S*3*<sup>m</sup>* and the magnetic flux density *B*<sup>3</sup> are:

**Figure 12.**

*(a) Schematic illustration of the laminated bilayer ME in L-T mode and local coordinates for the (b) piezoelectric and (c) piezomagnetic layer.*

*Equivalent Circuit Model of Magnetostrictive/Piezoelectric Laminated Composite DOI: http://dx.doi.org/10.5772/intechopen.107749*

$$S\_{3m} = s\_{33}^H T\_{3m} + d\_{33,m} H\_3 \tag{70}$$

$$B\_3 = d\_{33,m} T\_{3m} + \mu\_{33}^T H\_3 \tag{71}$$

Being *H*<sup>3</sup> the magnetic field intensity in the magnetostrictive layer along the zdirection, *T*3*<sup>m</sup>* the stress in the piezomagnetic layer along the x direction, *s H* <sup>33</sup> the elastic compliance to constant *H*, *d*33,*<sup>m</sup>* the piezomagnetic constant, and *μ<sup>T</sup>* <sup>33</sup> the magnetic permeability at constant stress.

Note that in the magnetostrictive layer, the local coordinate "3" (which always designates the direction of magnetization) coincides with the direction of movement, therefore all the subscripts are "3" in the constitutive equations of the magnetostrictive material. In this case, the phases operate as a magnetostrictive transducer in mode 33 and as a piezoelectric transducer in mode 31.

Considering mechanical dissipation, the expressions for the CME and DME coefficients near their resonance frequency became:

**Figure 13.**

∣*αCME*∣ *at frequencies from 20 kHz to 90 kHz, for ME laminate in L-T mode.*

**Figure 14.** ∣*αDME*∣ *at frequencies from 20 kHz to 90 kHz, for ME laminate in L-T mode.*

$$a\_{\rm CME(L-T)} = -\frac{\frac{d\_{\rm 33m}}{s\_{\rm 33}^T \mu\_{\rm 3}^T} \frac{d\_{\rm 33p}}{s\_{\rm 11}^T} r\_p}{\left(\frac{d\_{\rm 33m}}{s\_{\rm 33}^T \mu\_{\rm 3}^T}\right)^2 r\_m + \frac{\pi \overline{\rho}}{8 \mu\_{\rm 33}^T} \left(1 - \frac{\overline{\rho}^2}{\alpha\_r^2}\right) + \frac{j \pi \overline{\rho}}{8 \mu\_{\rm 33}^T Q\_{\rm mc}}}\tag{72}$$

$$a\_{\rm DME(L-T)} = \frac{\frac{d\_{33,m}}{s\_{33}^H \mu\_{33}^I} \frac{d\_{34,p}}{s\_{11}^P} r\_m}{\left(\frac{d\_{33,m}}{s\_{33}^H \mu\_{33}^I}\right)^2 r\_m + \frac{r\_p}{\overline{r\_{33}} \mu\_{33}^T} \left(\frac{d\_{31,p}}{s\_{11}^E}\right)^2 + \frac{\pi \overline{\rho}}{8 \mu\_{33}^T} \left(1 - \frac{\alpha^2}{\alpha\_r^2}\right) + \frac{j \pi \overline{\rho}}{8 \mu\_{33}^T Q\_{mc}}} \tag{73}$$

And the expression for the voltage coefficient in the ME laminate in L-T mode is calculated by means of Eq. (69).

Using Matlab® software to develop the programming codes for the simulation in L-T mode, the results obtained are shown in **Figures 13**–**17**. The approximation *s H* <sup>11</sup> � 3*s H* <sup>33</sup> was considered in the magnetostrictive layer [8].

**Figure 15.** *Maximum of* ∣*αCME*∣ *and* ∣*αDME*∣ *versus rm, for ME laminate in L-T mode.*

*CME and DME resonance frequencies (f resCME and fresDME) as a function of magnetostrictive phase content, for ME laminate in L-T mode.*

*Equivalent Circuit Model of Magnetostrictive/Piezoelectric Laminated Composite DOI: http://dx.doi.org/10.5772/intechopen.107749*

**Figure 17.** *α<sup>V</sup> versus magnetic phase content, in fresDME, for ME laminate in L-T mode.*

From the results obtained, it is highlighted that the range of resonance frequencies, both of CME and DME, present a shift towards higher frequencies for the laminate with L-T configuration: the range of resonance frequencies in L-T mode is Δ*f* � 120 � 226 *kHz* (**Figure 16**), while for a T–T configuration it is Δ*f* � 108 � 135 *kHz* (**Figure 10**).

Regarding the ME response analyzed through the voltage coefficient, for the L-T mode a maximum value of *α<sup>V</sup>* � 880*mV=Oe* is found corresponding to an optimal piezomagnetic phase content of *rm* ¼ 0*:*2 at a frequency *f* � 135 *kHz* (**Figure 17**).

For a better comparison between the results obtained in the L-T and T–T modes, **Figure 18** shows the voltage coefficients *α<sup>V</sup>* (near to resonance frequencies) with the magnetic phase content *rm*. It is observed that the responses of the material operating in L-T mode are greater for any magnetic phase content, with respect to the responses in T–T mode. This behavior may be due, at least in part, to energy issues of the ferrite phase. Due to the geometry of the material, the energy component of shape anisotropy presents a lower demagnetization factor in the L-T case, where the magnetization is

**Figure 18.** *Maximum α<sup>V</sup> versus rm of the ME laminates, in L-T and T–T modes.*

parallel to the length of the material (axis of greatest dimension). This agrees with the trend of the magnetic hysteresis cycles of pure ferrites in bulk, where with measurements in parallel configuration the magnetic responses are greater than those obtained in perpendicular configuration.

The results obtained by applying the circuit model to the barium titanate – nickel ferrite system (Sections 8 and 9) have similar trends to profiles reported by other researchers in analogous systems [14–16]. Additionally, the model was applied in a PZT–Terfenol system [9], observing similar trends to previously published results [17–19]. These similarities in trends allow us to infer that the designed circuit model is useful for modeling the behavior of ME laminated composites. It should be noted that the values of the CME and DME coefficients obtained in the simulation are expected to be somewhat higher than the real values, due to for example the fact that interface effects have not been considered in the model.

## **10. Conclusions**

From the development of magneto-mechano-electric equations, in this chapter it was possible to simulate the direct and converse ME effects of composite laminates by developing an equivalent circuit model, thus predicting the behavior of materials in the presence of a dynamic electromagnetic field.

It is found that the ME material has significantly higher CME and DME coefficients near the resonance frequency. The resonance frequency depends on mode of operation and the phase thicknesses.

For the T–T and L-T modes, the coefficients of the direct and converse effects become similar for materials in which the structures have similar phase content (*rm* � 0*:*5) suggesting that both effects share the same energy transmission capacity and are relative to the thickness ratio.

We found that the resonant frequency range of CME and DME exhibits a shift towards higher frequencies for the laminate with L-T configuration

(Δ*f* � 120 � 226 *kHz*) compared to T–T configuration (Δ*f* � 108 � 135 *kHz*). From the results obtained, it is observed that for the barium titanate – nickel ferrite system, the variation of T–T to an L-T mode generates notably higher values of *α<sup>V</sup>* coefficients, in addition to a shift towards higher values of resonance frequency. The maximum voltage coefficient for the T–T mode has a maximum value of *α<sup>V</sup>* � 80 *mV=Oe* with an optimal piezomagnetic phase content of *rm* ¼ 0*:*3 at *f* � 116 *kHz*. In the L-T mode, the voltage coefficient reaches a maximum value of *α<sup>V</sup>* � 880*mV=Oe* with an optimal piezomagnetic phase content of *rm* ¼ 0*:*2 at *f* � 135 *kHz*.

The results obtained allow us to infer that, at least in the frequency region studied, the tuning of these frequencies is an important step in the design of these materials for use as transducer devices.

*Equivalent Circuit Model of Magnetostrictive/Piezoelectric Laminated Composite DOI: http://dx.doi.org/10.5772/intechopen.107749*

## **Author details**

María Sol Ruiz\* and Adrián César Razzitte Faculty of Engineering, University of Buenos Aires, Ciudad Autónoma de Buenos Aires, Argentina

\*Address all correspondence to: mruiz@fi.uba.ar

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Section 2
