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

Holland, the term related to the piezoelectric coupling is proportional to EΠd", which must obviously possess units of energy density (subscripts are omitted for simplicity). In contrast, the piezoelectric hysteresis area is proportional to either the product of xE (converse effect) or DΠ (direct effect), none of which possess energy density units. A very nice experimental example of the reduction of the total power dissipation due to piezoelectric coupling was shown for Sm-doped PbTiO3 [4].

Coming back to our case on BFO, it must be emphasized at this point that the negative converse piezoelectric phase angle has also been confirmed in the direct piezoelectric response of BFO [91]. In addition, a negative phase has also been detected in the lattice microstrain response of BFO ceramics to external electric field,

#### **Figure 4.**

*(a) Scanning electron microscopy (SEM) image of the surface of BFO ceramics with visible grains and domains. At the lower part of the image, a grain was highlighted for clarity with the thick white line indicating the grain boundary and the thin red dashed lines indicating DWs. Note that the domains, which are separated by these DWs, are clearly seen as alternating dark/bright bands in the SEM image. The other two regions highlighted with blue circles were further analyzed by means of EBSD analysis. In each of the two regions, the type of DW (either 180° or non-180°) was determined based on the orientation of [111] polar axis of the rhombohedral lattice (with respect to the surface sample plane) in domains adjacent to the analyzed DW (for details, see text and reference [79]). (b) c-AFM maps of the same regions which are in panel (a) highlighted with blue circles. Irel on the scale bar signifies relative current, displayed with respect to the average background current signal. The bright lines observed at DWs (see blue arrows on the maps), which correspond to an increased local electric current signal, confirm the enhanced conductivity at both 180° (bottom c-AFM image) and non-180° DWs (top c-AFM image). Adjacent to the c-AFM maps are the electric current profiles measured by scanning the AFM tip on the surface along the white dashed lines noted on respective c-AFM maps. The peaks noted by blue arrows clearly emerge from the electric-current background, supporting the idea of electrically conducting 180° and non-180° DWs in BFO ceramics. Parts of the figure are reprinted from reference [79] with the permission of John Wiley and Sons.*

which was characterized by in situ XRD stroboscopic analysis [84]. This is perhaps the most important evidence of a negative piezoelectric phase angle measured in any piezoelectric so far because, prior to that work, such response (to the best of the author's knowledge) was demonstrated only on the level of macroscopic measurements.

As anticipated in the initial discussion of the results shown in **Figure 3**, the negative piezoelectric phase angle is a strong indication of M-W piezoelectric process that is very common in, e.g., piezoelectric composites [46, 47]. In analogy to the dielectric M-W relaxation, characterized by giant apparent dielectric permittivity [50], the modeling of Turik et al. showed the same effects in the piezoelectric M-W analogue [45]. As explained earlier, the M-W relaxation has origin in the internal electric field redistribution in a medium, where local regions exhibit different electrical conductivities; in the frame of modeling, such regions are often implemented in the form of M-W bilayer units, where the two layers are described by different conductivities [48, 51]. This is the reason behind a large piezoelectric M-W effect in some Aurivillius phases, such as Bi4Ti3O12 [48]. These ceramics, in fact, tend to show anisotropic microstructure with elongated plate-like grains that are characterized by different electrical conductivities in the direction parallel or perpendicular to the plane of the plates. The piezoelectric M-W effect in heterogeneous Bi4Ti3O12 ceramics is thus easy to support using arguments of anisotropic conductivity.

Unlike in Bi4Ti3O12, the M-W features observed in the piezoelectric response of BFO (**Figure 3**) are more difficult to interpret because significant anisotropy in the conductivity in a homogeneous perovskite oxide, such as BFO, is not expected, at least not to a level as in layered Aurivillius-type structures. Also, BFO is characterized by a microstructure typically consisting of equiaxial grains (as it is illustrated in **Figure 4** in the next section). It should be recalled, however, that the piezoelectric M-W effect in BFO is particular in that it shows, in addition to the negative piezoelectric phase angle (**Figure 3**), a very strong DW contribution observed only at low (sub-Hz) driving frequencies (**Figure 2c**). The overall data thus suggest a piezoelectric M-W effect provoked by DWs. The idea becomes reasonable when DWs are electrically conducting as the conductivity is what triggers the M-W effect. As it will be shown in the next section, it is exactly the conducting DWs that likely cause large anisotropy in the conductivity from grain to grain or across cluster of grains. Obviously, in BFO the situation is more complex than in other known piezoelectric M-W cases, simply because the features that are triggering the M-W effects, i.e., the DWs, can also move inside the grains under applied fields.
