**1. Introduction**

Historically speaking, probably the most important examples of dielectric and piezoelectric nonlinearity and hysteresis in ferroelectrics can be found already in the initial lines of the introductory section of the paper by Joseph Valasek from 1921 [1], which we use today to celebrate the 100th anniversary of the discovery of ferroelectricity [2]. Despite struggling with moisture-sensitive Rochelle salt crystals, Valasek finally managed to publish the first ever case of a ferroelectric hysteresis loop. Today, such hysteresis of the polarization response to the external electric field, originating from ferroelectric domain switching, is essential in, e.g., ferroelectric random-access

memories (FeRAMs) [3]. Hysteretic and nonlinear responses to external fields, however, are not only observed at switching conditions. Deviations from simple linear relationship between polarization (P) and electric-field (E) (dielectric response), P and mechanical stress (Π) (direct piezoelectricity) and mechanical strain (x) and E (converse piezoelectricity), are very common at subswitching driving conditions at which many important devices operate, such as capacitors, sensors and actuators. Such responses are measured in both bulk and thin-film ferroelectrics [4–6].

The origins of subswitching macroscopic nonlinearity and hysteresis alongside the frequency dependence of the property coefficients, which are directly relevant for device performance, are difficult to assess due to complex interrelated mechanisms often operating simultaneously over a wide length-scale range of the material. By risking of being too general, one could say that the major players of these mechanisms are domain walls (DWs) and similar interfaces [7], charged point defects [8–11], grain boundaries [12], spontaneous ferroelastic strains [13, 14], oxygen octahedra tilts [15] and secondary phases [16], if only the widely discussed examples are listed. Most of these features are always present in polycrystalline ferroelectric materials and are capable of cross-interacting via electric and/or elastic fields (a classical example that every ferroelectrician should be aware of is the DW-defect electro-elastic interaction [17, 18]). Complicating the picture is the fact that many of these features are strongly dependent on the synthesis conditions. Despite playing a key role, some of them, such as point defects, could not be directly visualized until recently [19–22], meaning that the interpretations of their role in the macroscopic response are supported by either indirect analysis or other reasoning (see next section). Without extensive knowledge in the area and with just a little intuition, it is pretty easy to understand why dielectric and piezoelectric nonlinearity as well as hysteresis in ferroelectric and similar materials are so difficult to understand and predict. Based on my personal opinion, the results shown in this chapter are the perfect example to illustrate the difficulties in elucidating the mechanisms responsible for the piezoelectric nonlinearity and hysteresis in a complex material, such as BiFeO3 (BFO). The results that I am going to present have been acquired over eight long years of multidisciplinary research focusing on theoretical and experimental studies and involving a large number of researchers from different fields.

One of the most important and widely-recognized origins of nonlinearity and hysteresis in the subswitching electrical and electromechanical response of ferroelectrics is the displacement of domain walls (DWs) [5, 23–26]. These are interfaces separating domain regions in which the spontaneous polarization is directed in one of the symmetry-allowed directions. Displacements of DWs are mechanistically treated via the interaction with charged points defects, which act as pinning sites and, depending on their type, mobility and location, affect the movement of DWs under applied fields (this issue will be thoroughly discussed in the next section). Reversible and irreversible motion of DWs in a ferroelectric with randomly distributed pinning sites (defects) is described by the Rayleigh law (RL), which is vastly used to quantify and cross-compare the subswitching responses of ferroelectric ceramics and thin films [6, 27]. For the converse piezoelectric effect (same is valid for the direct effect or dielectric response), the RL can be expressed by a set of two equations:

$$d'(E\_0) = d\_{init} + a \cdot E\_0 \tag{1}$$

$$\text{ax}(E) = (d\_{init} + a \cdot E\_0) \cdot E \pm \frac{a}{2} \left( E\_0^2 - E^2 \right) \tag{2}$$

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

For simplicity, the subscripts of the piezoelectric coefficient are omitted. Eq. (1) describes the linear dependence of the real component of the piezoelectric coefficient (*d*<sup>0</sup> ) on the electric-field amplitude (*E0*) with *dinit* and *α* representing the reversible and irreversible Rayleigh coefficients, respectively. Note that *dinit* corresponds to the coefficient extrapolated at zero field or *dinit* ¼ *d*<sup>0</sup> ð Þ *E*<sup>0</sup> ¼ 0 . Eq. (2) is a multivalued function and describes the strain (*x*) versus electric-field (*E*) relationship where the positive and negative signs, separating the first and the second term of the equation, are related to the strain branches defined by descending and ascending *E*, respectively. The second quadratic term of Eq. (2) thus defines the hysteresis. Note that *nonlinearity* refers here to the *nonlinear* relationship between strain (*x*) and electric field (*E*) (second quadratic term of Eq. (2)), in which case the proportionality coefficient, *d'*, is *linearly* dependent on the field amplitude (Eq. (1)).

By carefully inspecting the Rayleigh equations, it is not difficult to understand that the essence of this model is the intimate relationship between the nonlinearity and hysteresis. For example, in Eq. (1) the irreversible coefficient *α* defines the nonlinearity by the field dependence of the piezoelectric coefficient, however, *α* also appears in the second term of Eq. (2) describing the hysteresis (by the different upfield and down-field strain branches). Alternatively, one can easily realize the nonlinearity-hysteresis relation of RL in the second term of Eq. (2), which sets the nonlinear (quadratic) relation of strain to field while, at the same time, representing the hysteresis. In other words, hysteresis arises from nonlinearity and vice versa. A mathematical test is straightforward: in the absence of the irreversible contribution (*α* = 0), there are two important consequences: (i) the coefficient *d'* becomes independent on *E0* (Eq. (1) reduced to *d*<sup>0</sup> ð Þ¼ *E*<sup>0</sup> *dinit*), meaning that the response is linear (i.e., the coefficient is constant), and (ii) the hysteresis vanishes (Eq. (2) reduces to the first anhysteretic term). Therefore, every irreversible DW displacement contributes to nonlinearity and hysteresis in a unique way set by the RL [27].

When mechanistically reasonable, the irreversible coefficient *α*, experimentally extracted from the measurements of, e.g., longitudinal converse piezoelectric coefficient d33 as a function of field amplitude E0 (se Eq. (1)), is commonly used to quantify (irreversible) DW contribution to piezoelectric properties (an analogous approach is used to quantify DW contribution to dielectric permittivity). A rigorous test to find out whether the measured data can be approximated by RL is to at least validate the nonlinearity-hysteresis relationship implied by RL. This can be done by verifying whether the experimental hysteresis can be fitted by Eq. (2) using the coefficient *α* that is determined from the d33 versus E0 slope [6, 27]. Importantly, other parameters of the measured response can be checked against the RL predictions. For example, using the Fourier-series analysis of Rayleigh equations (derivations can be found in Ref. [27]) and the hysteresis-area analysis in complex mathematical formalism (see, e.g., references [24, 28]), it can be shown that the ratio between the increment in the real piezoelectric coefficient (Δ*d*<sup>0</sup> ¼ *d*<sup>0</sup> ð Þ� *E*<sup>0</sup> *dinit*) and the increment of the imaginary piezoelectric coefficient (Δ*d*<sup>00</sup> ¼ *d*00ð Þ� *E*<sup>0</sup> *d*00ð Þ *E*<sup>0</sup> ¼ 0 ) is constant (equal to <sup>4</sup> <sup>3</sup>*<sup>π</sup>*) and does not depend on the external field amplitude. This can be easily verified if the experimental piezoelectric coefficients and piezoelectric phase angles (or dielectric permittivity and losses) are known for different driving fields [14]. Other response parameters that can be evaluated against the RL predictions are embedded in the third harmonic response; this is discussed in detail in Ref. [29].

It has been shown that the subswitching response of a number of ferroelectric materials well obey the RL. Examples include donor-doped soft Pb(Zr,Ti)O3 (PZT) ceramics [5, 27, 28, 30], coarse-grained undoped BaTiO3 [31], high-Curie-temperature piezoceramics based on BiScO3-PbTiO3 [24], textured (K0.5Na0.5)NbO3 (KNN) ceramics [32], Aurivilius-type Nb-doped Bi4Ti3O12 [31], some relaxor-ferroelectric Pb (Mg1/3Nb2/3)O3-PbTiO3 compositions [33, 34] and Pb-based thin films [35, 36]. It has to be pointed out, however, that even in these cases, strictly speaking, a good match between the Rayleigh model and the measured subswitching response is typically observed in a limited driving field range, sometimes referred to as the "Rayleigh range" [5, 27]. This is expected, considering that RL is an idealization and assumes DW motion in a perfectly random pinning potential. It is clear that real materials' responses may come close to this situation; however, in most cases deviations from RL predictions are naturally observed and should not be surprising. It is legitimate to think that these deviations have microscopic origins different from DW motion. However, this is not necessarily the case. A nice supporting example can be found in soft PZT, which often exhibit sublinear, quasi-saturating field dependency of the piezoelectric coefficient at relatively weak fields, clearly deviating from the perfect linear behavior predicted by Eq. (1) [4]. In this case, Preisach approach becomes useful as it shows that such response can be understood by considering a non-uniform pinning potential where the concentration of weakly pinned DWs is higher than those that are strongly pinned (note that the two should be equal in a perfect Rayleigh random pinning potential, resulting in a flat Preisach distribution function) [37]. Therefore, while the experimental data deviates from the RL, in some cases this deviation can still be explained by DW motion without necessarily invoking other mechanisms unrelated to DWs.

Apart from those cases where the response is close to RL predictions, the subswitching response can be clearly non-Rayleigh. Acceptor-doped hard PZT [30, 38], Sm-doped PbTiO3 [27] or monoclinic Pb(Mg1/3Nb2/3)O3-PbTiO3 (PMN-PT) ceramic compositions with relaxor features [34] are just few of such examples. In this chapter, I am going to present another case that has attracted particular attention in recent times. The case is on BFO, a perovskite that is unique for at least two reasons: (i) because it contains electrically conducting DWs that are formed spontaneously and are attractive for applications in nanoelectronics [39] and (ii) due its high Curie temperature (835°C) [40, 41], which makes it the key perovskite in the development of novel compositions for high-temperature piezoelectric applications [42]. As it will be shown in the next section, enhanced conductivity at DWs completely alters the DW dynamics under applied subswitching fields. I will give evidence of a very large nonlinear contribution to the total piezoelectricity in BFO arising from irreversible DW dynamics, which is strongly dependent on the frequency of the external driving field. Along with a peculiar hysteretic behavior, characterized by a negative piezoelectric phase angle, and an unusual clock-wise rotational sense, I will strive to demonstrate how complex the dynamic response of electrically conducting DWs can be and how careful should we be in analyzing and treating materials that show unique behaviors. Somewhat surprisingly, I will show that despite the charged point defects, which are accumulated at DWs and are responsible for the local conductivity, act as pinning sites (as explained in Section 2), they can even increase the DW mobility and largely contribute to the enhanced nonlinear converse piezoelectric response. This may happen because the local conductive paths through the grains, set by the conductive DWs, lead to a redistribution of internal fields, effectively resulting in grains or grain families inside which the electric field is largely enhanced with respect to the field applied externally. The mechanism, called nonlinear Maxwell-Wagner piezoelectric effect (explained in Section 3), can be supported by simple

analytical modeling (Section 4) and should be more seriously considered when engineering BFO-based materials for high-temperature piezoelectric applications.
