## 2 Materials and Methods  
### Research on State-of-the-Art Piezoelectric Materials  
This section of the article reviews state-of-the-art man-made piezoelectric materials in terms of their material properties, which are considered as figures of merits (FOM)  
Figure 1: Potential piezoelectric actuators: (**a**) plates of the same dimensions/different materials and (**b**) plates of the same material/different dimensions. in piezoelectric actuator applications. The key figure of merit in piezoelectric actuator application areas is the piezoelectric strain coefficient \(d_{33}\) (longitudinal) or the piezoelectric stress coefficient \(\varepsilon_{33}\)[14]. Piezoelectric materials are intelligent materials that generate an electrical signal when subjected to mechanical effect, and mechanical strain when subjected to electrical effect. The first phenomenon is called the direct piezoelectric effect, which is the principle of operation of piezoelectric sensors and generators, while the second is the reverse piezoelectric effect, which is the principle of operation of piezoelectric actuators and motors. State-of-the-art synthetic piezoelectric materials can be categorized as single crystals, ceramics, polymers, composites, and piezoelectric thin films. Single Crystal Piezoelectric Materials. In the world of piezoelectric materials, well-known and high-performance single crystal piezoelectric materials are categorized under the lead-based relaxor-lead titanate (\(Pb(AB)O_{3}-PbTiO_{3}\)) class, where (\(A=Mg^{2+}\), \(Zn^{2+}\), \(Ni^{2+}\),..., \(B=Nb^{5+}\), \(Ta^{5+}\), \(W^{6+}\),..) [15]. The research carried out by [15; 16; 17; 18] showed that the high performance of the aforementioned class of materials is due to the presence of a morphotropic phase boundary (MPB) in the crystal. Two important systems in the relaxor-PT family are \(Pb(Mn_{1/2}Nb_{2/3})O_{3}\)-\(PbTiO_{3}\) (PMN-PT) and \(Pb(In_{1/2}Nb_{1/2})O_{3}\)-\(Pb(Mn_{1/2}Nb_{2/3})O_{3}\)-\(PbTiO_{3}\) (PIN-PMN-PT) [6]. Piezoelectric Ceramics. Lead-based ceramics have been found to exhibit excellent piezoelectric properties and are used in a variety of electronic devices. Particularly, lead zirconate titanate (\(Pb[Zr_{x}Ti_{1-x}]O_{3}\), \(0<x<1\)), more commonly known as PZT, and related compositions have been used in various sensors and actuators due to their excellent piezoelectric properties [6]. Extensive research has been carried out to optimize the piezoelectric properties of PZT using a variety of techniques. Of these techniques, the addition of various dopants or substitutional impurities has successfully tailored PZT ceramics to specific applications. Dopants can be donors or acceptors, depending on the characteristics of PZT to be optimized. The addition of donor dopants such as \(La^{+3}\), \(W^{+6}\), \(Nb^{+5}\), and \(Sb^{+5}\) increases piezoelectric coefficient, dielectric constant, and electromechanical coupling factor [6]. Such ceramics obtained by adding donor dopant are called soft PZT ceramics and are therefore appropriate for low frequency actuators [6]. Among soft PZT ceramics, PZT-5A, PZT-5J, and PZT-5H are the most widely used ceramics for actuator applications, and the respective piezoelectric strain coefficients \(d_{33}\) are 390, 485, and 650 \(pCN^{-1}\)[19]. The addition of acceptor dopants such as \(K^{+}\), \(Na^{+}\), \(Fe^{+3}\), \(Al^{+3}\), and \(Mn^{+3}\) decreases the piezoelectric coefficient, dielectric constant, and electromechanical coupling factor. PZT-4 and PZT-8 are commonly used hard PZT materials in sensors, and their piezoelectric strain coefficients are 372 and 375 \(pCN^{-1}\), respectively, as stated by [6; 19]. Lead-free piezoelectric ceramics. Due to the fact that lead is toxic to the environment and causes serious environmental pollution, the development of lead-free piezoelectric ceramics has attracted great interest.