**4. Analytical modeling of nonlinear Maxwell-Wagner piezoelectric relaxation arising from electrically conducting DWs**

While, pragmatically speaking, the idea of conducting DWs giving rise to macroscopic piezoelectric M-W effects does fit into a reasonable explanation, the details of the mechanism are rather complicated. The reason is the complex situation in a polycrystalline ceramic matrix containing randomly oriented grains, each characterized by a domain structure with different DWs and corresponding DW planes oriented in various directions with respect to the reference external field axis. The DW planes are assumed to provide conductive paths through individual grains or in local regions inside the grains, modifying the internal electric fields. If further grain-tograin elastic interactions are considered, which have a significant effect in polycrystalline piezoceramics [26], modeling may become extremely difficult. In the first

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

approximation, therefore, the model can be simplified and reduced to a level that allows the major macroscopic parameters to be predicted and compared with experimental data. As demonstrated for the case of direct piezoelectric response of Aurivillius phases [48], analytical bilayer modeling is sometimes sufficient. A similar case study on BFO, but additionally elaborated to incorporate nonlinear effects, will be presented in this section.

In the first place, it is important to work out a physical picture onto which to construct the model. Following the original discovery of conducting DWs in epitaxial BFO thin films [92], conduction at DWs in BFO polycrystalline samples have been determined using the same c-AFM measurements, which make it possible to probe the local conductivity of the sample. In BFO films, the epitaxial growth determines the crystallographic orientation of the film with respect to the substrate plane, so that the determination of the type of DWs in rhombohedral (R) BFO (71°, 109° or 180°) is facilitated. In ceramics, this is more difficult as the orientation of the grains is random and it is thus not known when imaging the surface with microscopy techniques. A simple way to accomplish the task is to determine the orientation of the polar [111] axis of the BFO R symmetry in adjacent domains in a selected grain using, e.g., electron back-scattered diffraction (EBSD) analysis. By indexing the Kikuchi patterns obtained in individual domains, as those highlighted in blue circles in **Figure 4a**, with the available R BFO space group, the angle between the [111] vectors in adjacent domains can be determined. Once these angles are known and without going into detailed geometrical analyses, which may be non-trivial (see the example in the supplementary material 1 of the paper [93]), a separation between 180° and non-180° DW-type is possible. In the former case, the angle should be zero ([111] vectors in adjacent domains are parallel), while in the second is non-zero. Note that further analysis is complicated by the fact that EBDS cannot determine the orientation of the ferroelectric spontaneous polarization, but only the ferroelastic distortion.

EBSD analysis was used to determine the 180° and non-180° DWs in the example shown in **Figure 4a** (see also reference [79] for further details). In the next step, these same regions were analyzed by c-AFM (**Figure 4b**) to finally confirm that both 180° and non-180° DWs in R BFO ceramics exhibit enhanced electrical conductivity with respect to that measured in the domains (see also the electric-current profiles in **Figure 4b** measured across DWs as indicated with the dashed white lines in respective c-AFM maps). The results are fully consistent with the atomic-scale microscopy analysis, which confirmed the presence of p-type charge carriers, identified as Fe4+ states, concentrated inside all three DW variants (71°, 109° and 180°) of BFO [19], resulting in the domains mostly likely depleted from the charges. The p-type conductive nature of the DWs is supported by annealing studies in controlled atmospheres, reported in the same paper, while the dynamic pinning effects of the p-type carriers is reported in Ref. [75].

The analysis of the effect of conductive DWs on the piezoelectric response can be reduced, in the first approximation, to a two-grain problem as shown in **Figure 5a** (left schematic). Since ceramics are composed of randomly oriented grains containing different type of DWs with various orientations, it is legitimate to analyze a hypothetical case of two grains oriented along [100] and [111] directions with respect to the externally applied electric field vector. The choice of these orientations will become evident in the subsequent discussion. If the two grains contain only 71° DWs percolating the grains, then the situation should closely resemble the left schematic shown in **Figure 5a** (detailed geometrical analysis of the DW angles in the two grains is reported elsewhere [84]). The key element of the model is the different orientation

#### **Figure 5.**

*(a) Schematic of the two-grain model (left) representing a serial bilayer M-W unit (right). The two grains are assumed [100]pc and [111]pc oriented with respect to field axis E and are denoted as grain 1 (red) and grain 2 (blue), respectively. Notation "pc" indicates "pseudocubic". The dashed lines inside the grains represent DWs. The DW orientation is assumed such that the DW planes in grain 1 are parallel to E, while those in grain 2 are not, forming an angle of 35° with respect to E axis. These angles should correspond to those of 71° DWs present in the two grains (for detail geometrical analysis, see reference [84]). Due to the different orientation of the electrically conductive DW in the two grains, the conductivity measured along E axis will be higher in grain 1 than in grain 2 (see σ<sup>1</sup> and σ<sup>2</sup> notation with upward and downward arrow in grain 1 and grain 2, respectively). This will result in the drop of the electric field in grain 1 (noted by down-arrowed E1) and rise of the electric field in grain 2 (uparrowed E2). The same difference in the orientation of the DW with respect to E will result in a lattice strain with no contribution from DW motion in grain 1 (see ΔL1), while displacements of DWs is expected in grain2, in addition to lattice strain (see larger ΔL2). To incorporate DW contributions, the calculations were performed assuming RL relations in grain 2 as described in detail in Ref. [94]. The two grains may be viewed as a M-W bilayer unit where each layer is assumed as a leaky piezoelectric element, characterized by its own piezoelectric coefficient di, dielectric permittivity εi, specific DC electrical conductivity σ<sup>i</sup> and volume fraction ν<sup>i</sup> (i = 100 or 111 indicating the two grain orientations). Panels (b–d) show driving-frequency dependent (b) internal electric field distribution (E1, E2), (c) piezoelectric microstrains (x1, x2) and (d) tangent of the piezoelectric phase angle (tanδ1, tanδ2) of the two grains (layers) as calculated from the model. The different curves in panels (b,c,d) correspond to different anisotropic parameter, i.e., ratio of the conductivities <sup>σ</sup>*<sup>1</sup> *σ*2 *, which were set to 1.01, 1.5, 2.5, 4 and 5 (the increasing tendency of <sup>σ</sup>*<sup>1</sup> *<sup>σ</sup>*<sup>2</sup> *is shown by arrows on individual plots and the corresponding curves are drawn with different lines, each corresponding to specific <sup>σ</sup>*<sup>1</sup> *<sup>σ</sup>*<sup>2</sup> *ratio as noted in panel b). The data in panels (b,c) were calculated for 16 kV/cm of externally applied sinusoidal field amplitude, while those shown in panel (d) were calculated for zero-field amplitude (linear piezoelectric tanδinit). (e) Effective (total) piezoelectric coefficient of the M-W bilayer unit normalized to the value at zero field (d33eff norm) as a function of external (nominal) electric field amplitude calculated for different driving field frequencies (the anisotropic parameter <sup>σ</sup>*<sup>1</sup> *<sup>σ</sup>*<sup>2</sup> *was fixed to a value of 5). Parts of the figure are reprinted from Ref. [94] with the permission of AIP Publishing.*

of conductive DWs in the two grains with respect to the external field axis, i.e., in the top grain 1 (red) the DWs are parallel to E, while in the bottom grain 2 (blue) they form a different angle. The net result is that the conductivity, measured along E, of grain 1 should be higher than that of grain 2 because in the former the charges may migrate along the vertical conductive DWs (see notations σ<sup>1</sup> and σ<sup>2</sup> in **Figure 5a** indicating the conductivity measured vertically in grain 1 and 2, respectively). Using the same reasoning, the conductivity of grain 2 should be higher when measured along a direction away from the E axis (not parallel). It is this anisotropy in the electric conductivity that it is assumed here to lead to M-W-like internal field redistribution (see down-arrowed E1 and up-arrowed E2 notations in **Figure 5a**, denoting the

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

internal fields). Note also that the bottom grain 2 is oriented in a way to give rise to a stronger DW contribution than the upper grain 1; in the ideal case, due to orientational constraints, grain 1 should exhibit only lattice strain as a response to the external E (in **Figure 5a** the microstrains arising from the two grains due to field application are noted as ΔL1 and ΔL2). Further details regarding the model and mathematical derivations can be found in the supplementary Section 5 of the paper by Makarovic et al. [94].

The assembly of the two grains represents a basic M-W bilayer unit. As in classical dielectric M-W modeling [50], the picture can be rationalized in terms of an equivalent circuit consisting of two leaky capacitors connected in series, where each leaky capacitor may be represented by an ideal capacitor and an ideal resistor connected in parallel (**Figure 5a**, right-hand schematic). Obviously, the piezoelectric effect is added to the two layers (grains). Due to the expected DW contribution in the bottom grain 2, the RL relations (Eqs. (1) and (2)) were used to calculate the piezoelectric response of this grain. In contrast, grain 1 is assumed to respond via intrinsic lattice piezoelectric effect, so its response can be modeled by the linear constitutive piezoelectric equation. For a set of dielectric, piezoelectric and conductivity parameters, which are within the margins reasonable for BFO (see reference [94]), the results of the modeling are shown in **Figure 5b**–**e**.

Driven by the anisotropy in the electrical conductivity, defined as the conductivity ratio of the two grains (*<sup>σ</sup>*<sup>1</sup> *σ*2 ), the internal fields in the grains (E1, E2) will be redistributed as a function of driving field frequency (**Figure 5b**). In accordance to the conductive behavior related to the different DW orientation in the two grains, the sinusoidal electric field applied externally to the bilayer serial structure will be redistributed at low driving frequencies (<10 Hz) such that the electric field in the top grain 1 (red color coding) will be reduced with respect to the nominal (external) field, while that of the bottom grain 2 (blue) will be increased instead (**Figure 5b**). In other words, due to leakage in grain 1 caused by the vertical orientation of conductive DWs, the field inside this grain will drop and will be thus transferred from the leaky grain 1 to the less-leaky grain 2 (see **Figure 5a**). This will happen at low driving frequencies as such driving conditions provide to the charges sufficient time to migrate along conductive DWs. Being proportional to the internal field, the microstrains of the two grains (**Figure 5c**) will show the same frequency behavior as that of respective internal fields, leading to either retardation and a peak in the positive piezoelectric phase angle (see tanδ<sup>2</sup> in **Figure 5d**), or relaxation and a peak in the negative piezoelectric phase angle (see tanδ<sup>1</sup> in **Figure 5d**). This M-W mechanism will ultimately lead to nonlinear effects: the increased internal electric field in grain 2 (E2 in **Figure 5b**) will boost the DW motion in this grain, resulting in effective nonlinearity enhanced at low driving frequencies as shown in **Figure 5e** (compare also with experimental data, **Figure 2c**). The piezoelectric M-W effect has thus a nonlinear character or, in other words, the piezoelectric nonlinearity (i.e., DW motion) is boosted due to the electrical conductivity.

At this stage, I have to point out that the presented model used to understand the experimental data on BFO (as those shown in **Figures 2c** and **3**) is largely simplified: (i) it considers only two isolated and unconstrained grains connected in series, neglecting the true elastic and electric boundary conditions of the analyzed grains set by the presence of other surrounding grains, (ii) it does not consider elastic coupling of the two isolated grains and transverse piezoelectric effects (as was done in, e.g., reference [51]), (iii) it assumes very simple domain structure in the two grains and

only one type of DW (i.e., 71°), and (iv) the conductivity of the grains set by the conductive paths along DWs is assumed to be fixed, although a dynamic conductivity is not unreasonable considering that the conductive DWs may switch locally (through a RL-like irreversible jump, for example), resulting in a modified conductive path through the switched DW. Despite all these limitations, however, it is surprising to realize that the simple two-grain model predicts all the key experimental observations. First, the model predicts both retardation and relaxation processes (**Figure 5c** and **d**), which are likely convoluted in the experimental response (as discussed for **Figure 3a**). Second, the model shows that the lattice strain response should exhibit a negative phase with respect to sinusoidal external field, thus leading the field signal (**Figure 5d**). Not only is this consistent with macroscopically measured piezoelectric strain leading the field signal (**Figure 3c**), but this is also exactly the behavior of the lattice strain deconvoluted from the total converse piezoelectric response of BFO ceramics using synchrotron XRD measurements [84]. Third, the important outcome of the model related to the low-frequency nonlinearity is a natural consequence of introducing RL relations, lining up with the macroscopic experimental data (**Figure 2c**) and the low-frequency DW contribution determined by synchrotron XRD measurements [84].

The low-frequency nonlinearity has recently been shown to be a response parameter that can be controlled by designing the fraction of conductive DWs in BFO ceramics via doping [94]. The idea was triggered by the outcomes of the model itself, further reinforcing the value and importance of simple modeling. It is probably needless to say that controlling nonlinearity and hysteresis is very important for the development of high-temperature piezoceramics based on BFO. This is indeed supported by a study on BFO showing that the temperature dependent piezoelectric response of these ceramics is strictly controlled by the same M-W processes described in this chapter [95]. Importantly, it was shown that the strong temperature dependence of the piezoelectric nonlinearity and hysteresis, which is of a direct relevance for the device operation, has origin in the thermally activated nature of the local electrical conductivity in BFO ceramics.

Simple analytical modeling was found to be a promising first step toward understanding the complex piezoelectric behavior of BFO arising from conductive DWs. It could be interesting, however, to model a more complex situation to account for the many different parameters that are necessarily neglected in the simple analytical approach. As shown in **Figure 5b**–**d**, the effect of the anisotropic parameter is crucial as it determines the strength of the M-W effect. For more advanced modeling, this parameter could be viewed as varying from one M-W unit to another in a complex ceramic matrix composed of interacting grains forming units with different time constants (τ is proportional to the ratio of the weighted sum of permittivity and conductivity of the individual layers in the bilayer unit [48, 84]). This could eventually account for the different regions inside the ceramics exhibiting different levels of electric-field redistribution (as shown by the different curves in **Figure 5b**). 3-dimensional (3D) finite element modeling based on a phenomenological approach has been recently demonstrated to be a powerful tool in predicting local electric fields in 3D ceramic matrices with defined porosity [96]. For solving complex problems, such as those encountered in elastically and electrically coupled grains in ferroelectric ceramics, multiscale modeling approaches show great promises [97–99].

Another interesting point that could be considered in the future is to push the relaxation to higher frequencies (equal to decreasing the τ value). This would make it possible to use the large response in a higher frequency range that is more relevant for

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

piezoelectric applications (in the example shown in **Figure 5**, τ = 0.13 s, corresponding to a relaxation frequency of frelax = 1.2 Hz). As discussed in the preceding paragraph, the anisotropy in the conductivity is also an important factor in tailoring the usable frequency range. It is thus tempting to consider designing a matrix with charged DWs, which can possess metallic-like conductivity, as recently demonstrated for BaTiO3 [100]. In this case, the DW conductivity may exceed the bulk conductivity for impressive 8–10 orders of magnitude, perhaps providing an opportunity in designing piezoelectric properties.
