# Dielectric Spectroscopy Studies and Modelling of Piezoelectric Properties of Multiferroic Ceramics
## 1 Introduction  
From a scientific point of view, multiferroic materials are a somewhat mysterious yet, at the same time, extremely interesting class of materials. Thanks to the unique properties of these materials, which are characteristic of both ferroelectric and ferromagnetic materials, they have a wide range of potential applications. Multiferroic materials can open newhorizons in the design of modern devices, especially those whose operation is based on controlling the electrical properties of materials (and thus influencing the parameters of the electrical circuit) by changing the magnetic field and vice versa, i.e., controlling the magnetic properties of materials by changing electric field strength [1]. Let us consider the terminal elements of bismuth ferrite (BiFeO\({}_{3}\))--barium titanate (BaTiO\({}_{3}\)) system. It is well known that BiFeO\({}_{3}\) has an ABO\({}_{3}\)-type perovskite structure and is one of the few materials characterised by the coexistence of ferroelectric and (anti)ferromagnetic properties at room temperature. Because of this essential feature of multiferroics, BiFeO\({}_{3}\) (BFO) is considered the most suitable among all available multiferroic materials for the development of practical spintronic devices and related technologies [2]. BaTiO\({}_{3}\) (BT) belongs to the corner-sharing oxygen octahedral material family (structure), which was discovered during World War II. Since then, it has remained one of the most widely used and systematically studied non-linear dielectric materials and is considered the prototype for ferroelectric ceramics as well as being the first piezoelectric ceramics transducer ever developed [3]. Since the discovery of BT in the early 1940s, it has been continually used in new industrial and commercial applications. Its unique physical characteristics, e.g., values of piezoelectric (\(d_{33}\sim\)190 pC/N) and dielectric (\(\epsilon_{r}\sim\)1700 and low dielectric losses at room temperature) parameters, enable the use of BT ceramics for capacitors, multilayer capacitors, ultrasonic transducers, pyroelectric infrared sensors, positive temperature coefficient resistors, and energy storage devices [4; 5]. However, neither BFO nor BT are actually used in their pure chemical form. To optimise properties for specific applications, they are often combined with various additives that adjust and enhance their basic properties. They are also combined with each other to form either compounds or ceramic-ceramic composites, which have received a lot of scientific interest [6; 7; 8]. To justify the above-mentioned points, let us point out a bulk ternary BiFeO\({}_{3}\)-BaTiO\({}_{3}\) and PbTiO\({}_{3}\) system [9] that was fabricated and studied as a potential high-temperature piezoelectric ceramics transducer. In case of Tm-doped 0.7 Bi\({}_{1-x}\)Tm\({}_{x}\)FeO\({}_{3}\)-0.3 BaTiO\({}_{3}\) (\(x=0\)-0.05) ceramics, it was found that Tm-induced structural modifications yielded improvements in the dielectric constant, remnant polarization, and remnant magnetization values [10]. On the other hand, studies of Mn-modified (\(1-x\))BiFeO\({}_{3}\)-xBaTiO\({}_{3}\) ceramics showed that the addition of Mn improved dielectric losses and increased dc resistivity in bulk BiFeO\({}_{3}\)-BaTiO\({}_{3}\), greatly enhancing the piezoelectric response; however, the poor stability of the poled domain structure caused rapid ageing and a large difference between low- and high-field piezoelectric charge coefficient \(d_{33}\)[11]. The main motivation to carry out the present research was the high potential for practical application that the materials of the BiFeO\({}_{3}\)-BaTiO\({}_{3}\) system have. In addition, the issues of the physics and technology of multiferroic materials are extremely rich, interesting, and non-trivial from a scientific point of view. Novel materials of rich functionality can be created by means of combining and properly modifying bismuth ferrite and barium titanate.