**2. Dynamic interactions between charged point defects and domain walls: a microscopic view**

In this section, we will take a look at idealized microscopic pictures describing the interactions between DWs and charged point defects, which have profound implications in the DW dynamics and play a key role in the macroscopic dielectric and piezoelectric nonlinearity and hysteresis. Before discussing these interactions, however, it is timely to emphasize few important points. First, the reader should be aware that DW dynamics is not the only mechanism giving rise to electrical and electromechanical hysteresis. An example that should be familiar to every scientist and engineer working in the field of dielectrics or capacitors, is the DC electrical conductivity, which can directly contribute to the imaginary dielectric coefficient and thus dielectric losses and P-E hysteresis. In ionic compounds, electron and holes tend to be trapped at local sites in the lattice; they locally polarize and deform the lattice, creating the so-called polarons. Being self-trapped, these charges can move via an activated hopping process, similar to ionic conductivity [43]. Interestingly, in both physical and mathematical sense, hopping charge transitions are indistinguishable from pure dipole reorientations, meaning that hopping conductivity can contribute not only to losses (imaginary dielectric permittivity) but also to polarization (real dielectric permittivity). Mathematical descriptions of these problems are given in the book by Jonscher, which is highly recommended for learning conductivity phenomena in dielectrics [44].

Second, the effect of the electrical conductivity can go beyond a simple contribution to the complex dielectric permittivity and can strongly affect piezoelectricity, too. This case, which is less trivial and somewhat difficult to digest (even for experts in the field), is the Maxwell-Wagner (M-W) piezoelectric relaxation [45–49]. As an analogue of the well-known dielectric M-W relaxation [50], it originates, generally speaking, from electrically conductive paths present locally inside the material, which lead to internal electric-field redistribution as a function of the driving field frequency and, consequently, to electromechanical relaxation phenomena. The effect was predicted for the direct [4, 48] and converse piezoelectric response [51], and even for purely elastic response [52]. It was experimentally observed in heterogeneous ceramic materials [48], two-phase polymer systems [53] and piezoelectric-polymer composites [47]. The piezoelectric M-W effect is described in detail in an older book on hysteresis [4], to which the reader should refer to as a support to the data presented in this chapter where the focus is on BFO ceramics.

To explain the dielectric M-W effect, either grain boundaries or sample-electrode interfaces are commonly considered, as these are the regions where depletion layers are likely to be formed, giving rise to inhomogeneous field distribution inside the material and, consequently, to M-W relaxation [50]. In this chapter, I will present a different case where local conductive paths originate from charges concentrated at DWs. It is intuitively understood that this case must be unique because, unlike grain boundaries or sample-electrode interfaces, the conductive paths provided by the DWs are dynamic: they can move and locally displace under external fields. One may simplistically view this scenario as a local conductivity that is not spatially confined.

The third point to consider is rather general. While the discussion in this chapter is mostly about DWs, the reader should understand that also the dynamics of other types of interfaces may lead to the same macroscopic effects associated with subswitching nonlinearity and hysteresis. As an example, those other interfaces may be represented by boundaries separating regions consisting of phases with different crystallographic symmetries, which are present in morphotropic compositions [7]. The role of interface boundary motion during subcoercive loading of morphotropic BiScO3-PbTiO3 ceramics was recently demonstrated by in situ X-ray diffraction (XRD) analysis [54].

We come back now to our case on DWs. In analogy to the well-known pinning of DWs in ferromagnetic materials [55], the same concept is used for ferroelectrics [56]. An important difference is that, on average, ferroelectric DWs are an order of magnitude thinner than ferromagnetic DWs [39, 57–59]. While it is clear that the subject is complex and cannot be reduced to few (over)simplified pictures, like those shown in **Figure 1**, it is surprising to realize that the behavior of many ferroelectric materials under applied fields can be qualitatively interpreted by considering relatively simple interactions between DWs and charged point defects. This is particularly true for high-field P-E (switching) hysteresis (see, e.g., reference [18]) and, to some degree, for subswitching responses (see, e.g., reference [4]). Precaution is, however, necessary because there are various parameters to consider, such as type, concentration, location and mobility of defects along with their binding state, as well as the presence of other pinning centers, including grain boundaries [12] or oxygen octahedra tilts [15]. Nevertheless, a simple exercise linking possible DW-defect interactions with the macroscopic response, which will be explained next, is very useful to assess the behavior of a given ferroelectric material.

Early studies identified three microscopic scenarios to describe the strong pinning and DW stabilization effects observed macroscopically by pinched and/or biased high-field P-E hysteresis loops of acceptor-doped PZT and BaTiO3 [8–11, 17]. The three scenarios are illustrated in **Figure 1a** and assume different locations of the pinning centers: (i) inside the domains (called "volume" or, sometimes, "bulk" effect), (ii) in DW regions ("domain wall" effect) and (iii) in or close grain boundary regions ("grain boundary" effect). The common feature of all these scenarios is that defects are arranged in a somewhat ordered manner and, by that, they stabilize the domain configuration via electric and elastic coupling [11, 18], inhibiting domain wall motion. Ordered defects are a characteristic of the so-called "hard" ferroelectrics, exemplified by acceptor-doped PZT. In contrast, "soft"

#### **Figure 1.**

*Simplified microscopic pictures of interactions between charged point defects and domain walls in (a) hard and (b) soft ferroelectrics. Ps and Pd are spontaneous and defect polarization, respectively. Thick black lines denote DWs and circles noted with + and – signs represent individual charged point defects (see also legend on the righthand side of the schematics). To be further noted is that Pd represents the well-known acceptor–oxygen-vacancy defect complexes (see text for details). The pictures shown in panel (a) were adapted according to the pioneering studies by Jonker [10], Carl and Hardtl [9], Lambeck and Jonker [8] and Robels and Arlt [11]. The scenario in panel (b) assumes disordered defects in soft materials as discussed in, e.g., reference [4].*

*Piezoelectric Nonlinearity and Hysteresis Arising from Dynamics of Electrically… DOI: http://dx.doi.org/10.5772/intechopen.98721*

ferroelectrics, represented by donor-doped PZT, are usually described by assuming disordered defects as shown in **Figure 1b**. Hard-soft transitions, induced in hard-type compositions by disordering the otherwise ordered defects, can be achieved by either electricfield cycling [9, 60] or quenching [30, 61, 62].

The "volume scenario" shown in **Figure 1a** (left schematic), which assumes binding of defects into complexes (see black arrows), is probably the only DW-pinning mechanism that is highly accepted in our research community and is widely used to explain hardening and aging in acceptor-doped PZT, BaTiO3 and similar perovskites (an excellent review on this topic can be found in the paper by Genenko et al. [63]). DW pinning effects mediated by orientation of defect complexes have been put in a theoretical framework more than 30 years ago [17]. More recent studies confirmed the formation of defect complexes and their orientation kinetics both theoretically (by first principles, [64]) and experimentally. In particular, electron paramagnetic resonance (EPR) spectroscopy was found to be a powerful tool due to its high sensitivity, not only to the binding state of certain defects (for details see [65]), but also to the orientation of the defect complexes in the material [65, 66], allowing thus to directly monitor the alignment kinetics of the complexes under the applied field [67]. In most cases, the identified complexes are of the acceptor–oxygen-vacancy type. The literature contains many examples of lead-based and lead-free ferroelectric compositions where such complexes have been identified by EPR; the list includes (but is probably not limited to) PbTiO3, PbZrO3, PZT, BaTiO3, KNN and (Bi,Na)TiO3 (BNT). The dopants in those cases are typical B-site acceptors, such as Fe, Cu and Mn [65, 66, 68–71].

The "DW scenario", which assumes defects accumulated in DW regions (**Figure 1a**, middle schematic), was first predicted by Postnikov more than 50 years ago [72]. Note that the original model does not assume defects bound into complexes. Interestingly, despite being seriously discussed in studies on fatigue mechanisms [73], the DW scenario received strong criticism (to be even labeled as "a well-known speculation") from those that were strictly in favor of the volume scenario [74]. Nevertheless, very recent studies on BFO ceramics using atomic-resolution electron microscopy, which will be discussed later, support the idea of this scenario and indeed suggest that is active in this material [75].

The "grain-boundary scenario" (**Figure 1a**, right schematic), often referred to as the space-charge mechanism, was extensively discussed in a combined theoretical and experimental frame against the volume scenario originally proposed by Arlt et al. [17]. Key hardening characteristics of acceptor (Fe) doped PZT, such as the dopant-concentration dependent aging time, were shown to be well predicted by the space charge model [76, 77]. A perhaps interesting observation is that phenomenological calculations revealed that the electrostatic pinning effects on DWs from the space charges can be two orders of magnitude stronger than the pinning effects provided by the aligned defect complexes (for the same charge-carrier concentration) [76]. Considering that grain boundaries are strained areas and sources of polarization discontinuities where point defects may be attracted due to electrostatic and elastic driving forces, the grain boundary scenario should probably be always considered when interpreting the macroscopic responses of ceramics to external driving fields, particularly if the ceramics are fine-grained.

It is clear that more than one of the three domain-stabilizing effects shown in **Figure 1a** may be active in a given material. Actually, the reason for considering only a single scenario is in the human nature seeking for the simplest explanation. A number of arguments supports the idea of multiple pinning mechanisms. I give two examples. In hard PZT, it was predicted by first principles that the acceptor–oxygen-vacancy defect complexes should energetically prefer to be situated closer to DWs (as shown

schematically with black arrows in **Figure 1a**, left schematic) [78]. This means that the pinning mechanism may consist of a combination of volume and DW effects. In another study based on conductive atomic-force microscopy (c-AFM) analyses, it has been shown that in BFO ceramics both DWs and grain boundaries exhibit enhanced electrical conductivity with respect to the bulk conductivity measured in the grain interiors [79]. This suggests the presence of mobile charges at both locations, which may act as pinning centers. In this same material, defect complexes based on oxygen vacancies have been also identified by EPR [80]. The overall data, therefore, point to an extremely complex situation where all three pinning scenarios shown in **Figure 1a** may be active. The challenge is to find whether and which mechanism is dominant. At present, conductive DWs appear to be the key features affecting the piezoelectric response of BFO, which will be discussed in detail in the next section.

In contrast to ordered defects in hard ferroelectrics, disordered defects in soft counterparts, as shown schematically in **Figure 1b**, are more difficult to be directly identified. Nevertheless, in the case of PZT, for example, the presence of disorder in donor-doped samples (in a pragmatic sense) is supported by the fact that the measured macroscopic subswitching nonlinearity and hysteresis are well consistent with the major predictions from the Rayleigh model [27]. Another argument is that EPR data on Gddoped PbTiO3 indicate no binding of the donor (Gd) dopant with the expected compensating Pb vacancies [81], supporting the results of first-principles studies on the same perovskite [78]. In addition, none of the two defects (donor dopants and Pb vacancies) are mobile below the Curie temperature of the material, where the defects can in principle be ordered because they are provided by electric and elastic driving forces originating from the spontaneous polarization and strain, respectively (see, e.g., the graphic explanations by Ren [18]). If the two defects are not mobile nor they show tendency of binding, then a "frozen-in" defect disorder state in donor-doped PT or PZT can be envisioned. It is not unusual, however, that the dopants segregate at the grain boundaries (a nice recent example is shown for Ti-doped BFO in Ref. [82]). The issue of defect segregation at or close to grain boundary regions should be more seriously considered, as recently pointed out by Slouka et al. [83].

Other discussions related to the nature of defects in soft PZT point to the likely possibility that the donor dopant reduces the concentration of oxygen vacancies in PZT (due to charge compensation), leading to a progressive transition from a state characterized by ordered defects (undoped PZT) to a state governed by disordered defects, as the donor dopant is added to PZT [29]. In this sense, experimental data even indicate that the dominant pinning centers affecting DW displacements and, consequently, nonlinearity, in both hard and soft PZT, could be oxygen vacancies. The lesson that can be learned from all these data is that the defect arrangements shown in **Figure 1** should not be treated individually; as a matter of fact, all of them may be present, to some degree, in a given ferroelectric. Finally, it has to be emphasized that the situation in soft materials is more complex, and the true origins of softening are still not clear (if the reader is interested in these issues, it is recommended to consult the work of Dragan Damjanovic [29, 78]).
