**3. Piezoelectric nonlinearity and hysteretic response of BFO ceramics: low-frequency nonlinearity and negative piezoelectric phase angle**

One of the distinct characteristics of the piezoelectric response of BFO is the strong dependence of the nonlinearity on the frequency of the driving field. This is illustrated *Piezoelectric Nonlinearity and Hysteresis Arising from Dynamics of Electrically… DOI: http://dx.doi.org/10.5772/intechopen.98721*

in **Figure 2**, which compares the converse piezoelectric response of BFO with that of hard and soft PZT. The PZT compositions correspond to the rhombohedral symmetry and were selected exactly to enable a comparison with the isostructural rhombohedral BFO, although symmetry is probably the only common aspect of these two perovskites. In the first place, it should be mentioned that the increasing tendency of d33 with the electric-field amplitude E0 observed in **Figure 2a**–**c** for the two PZTs (at all measured frequencies) and BFO (mostly at 1 and 0.2 Hz) is accompanied by an increase in the piezoelectric tanδ (not shown here). This suggests irreversibility in the response as implied by the intimate nonlinearity-hysteresis relationship predicted by RL (see Eqs. (1) and (2)). The nonlinear behavior in these samples can be thus related to irreversible domain wall contribution. This is supported by in situ XRD analysis on BFO [84] and similar PZT samples to those shown here [26].

As expected and described in the previous section, due to stronger DW pinning effects, hard PZT shows a lower absolute d33 value, which is less dependent on the field magnitude (**Figure 2a**) than that of soft PZT (**Figure 2b**). Also, the d33 field

#### **Figure 2.**

*Converse piezoelectric d33 coefficient as a function of electric-field amplitude E0 measured at different frequencies for (a) 1% Fe doped Pb(Zr0.6Ti0.4)O3 (hard PZT), (b) 1% Nb doped Pb(Zr0.6Ti0.4)O3 (soft PZT) and (c) BFO ceramics. (d) Comparison of the fraction of the piezoelectric d33 coefficient related to the irreversible contribution (XIR; see Eq. (3)) among the three samples. The response is in this case shown for two limited frequencies (90 or 100 Hz and 0.2 Hz) and as a function of relative electric field, i.e., E0/EC (EC is the coercive field determined by standard P-E hysteresis measurements, which are not shown here). For clarity, color coding of the curves is the same in all panels. The green arrow in panel (d) illustrates the strong effect of the driving electric-field frequency on the nonlinearity in BFO, which is absent in hard and soft PZT. The data correspond to fine-grained BFO sample (average grain size 1.8 μm) as reported in Ref. [79].*

dependency in soft PZT is sublinear with respect to the rather linear dependency in hard PZT. This behavior is retained at all frequencies used in the measurements. The sublinear trend in soft PZT has been discussed to some degree in the introduction of this chapter in relation to non-uniform pinning potential.

The next information can be obtained by quantifying the data. The slope of the d33-vs-E0 curves is for hard PZT �0.2 � <sup>10</sup>�<sup>16</sup> <sup>m</sup><sup>2</sup> <sup>V</sup>�<sup>2</sup> and is practically independent on the frequency. By contrast, in soft PZT, the slope (in this case calculated for 1 kV cm�<sup>1</sup> , which corresponds to 10% of coercive field EC, i.e., E/EC = 0.1) is an order of magnitude higher, i.e., 8.1 � <sup>10</sup>�<sup>16</sup> <sup>m</sup><sup>2</sup> <sup>V</sup>�<sup>2</sup> at 90 Hz, and further increases to 9.9 � <sup>10</sup>�<sup>16</sup> <sup>m</sup><sup>2</sup> <sup>V</sup>�<sup>2</sup> at 0.2 Hz. The frequency dependence of the irreversible coefficient of soft PZT has been reported earlier and correlated with interface pinning in a disordered medium [85].

As easily assessed from **Figure 2c**, the nonlinearity and the associated irreversible contribution to piezoelectricity in BFO is completely different than in PZT. Here, the nonlinear behavior is strongly dependent on frequency: the d33-vs-E0 slope is nearly zero at 100 Hz, precisely 0.003�10�<sup>16</sup> <sup>m</sup><sup>2</sup> <sup>V</sup>�<sup>2</sup> , and increases to 0.27�10�<sup>16</sup> <sup>m</sup><sup>2</sup> <sup>V</sup>�<sup>2</sup> when the frequency is reduced to 0.2 Hz (as in the previous case on PZT, the slope was calculated for the relative field E/EC = 0.1). Such a dramatic two-orders-ofmagnitude increase in the nonlinearity coefficient with reduced frequency is obviously absent in the two PZT variants.

A direct comparison between the nonlinearity of BFO and PZT is shown in **Figure 2d**. As a quantitative measure, we use the fraction of the total piezoelectric coefficient that is due to irreversible DW displacements (*XIR*). This parameter is essentially represented by the fraction of the field-dependent coefficient and can be derived from Eq. (1) as:

$$X\_{IR}(E\_0) = \frac{d\_{33}(E\_0) - d\_{33}(E\_0 = \mathbf{0})}{d(E\_0)}\tag{3}$$

where *d*33ð Þ *E*<sup>0</sup> and *d*33ð Þ *E*<sup>0</sup> ¼ 0 is the coefficient at a given field amplitude E0 and at zero field amplitude, respectively. As anticipated, the irreversible contribution in hard and soft PZT is practically independent on the frequency (**Figure 2d**, blue and red data). *XIR* reaches a maximum of 10% and 55% in hard and soft PZT, respectively. Interestingly, the irreversible contribution in BFO at 100 Hz is smaller than that of hard PZT and becomes higher at 0.2 Hz than that of soft PZT (for the same relative field E/EC; see green data in **Figure 2d**). Note that the large contribution from the displacements of non-180° domain walls at sub-Hz driving frequencies in BFO was recently confirmed by in situ XRD stroboscopic analysis [84]. One could imagine BFO as very hard material at high frequencies and very soft at low frequencies. The data, therefore, present a distinct hard-to-soft transition in BFO induced by the driving frequency. As will be explained throughout the rest of the chapter, this transition originates from the presence and dynamics of conductive DWs.

Another distinct feature of the piezoelectric response of BFO is the negative piezoelectric phase angle (here and in subsequent discussion, the phase is represented as the tangent of the piezoelectric phase angle, tanδp). This rather unusual response is particularly strong in coarse-grained BFO and is presented in **Figure 3**. From these data it is clear that the piezoelectric response of BFO is rather complex and show strong frequency dependence (**Figure 3a**). When inspected from high to low frequencies (right to left), the response can be described as a sequence of increasing and decreasing d33 in the frequency range 1–200 Hz and 0.1–1 Hz, respectively, giving *Piezoelectric Nonlinearity and Hysteresis Arising from Dynamics of Electrically… DOI: http://dx.doi.org/10.5772/intechopen.98721*

#### **Figure 3.**

*(a) Piezoelectric coefficient (d33) and tangent of the piezoelectric phase angle (tanδp) as a function of driving field frequency for coarse-grained BFO. The data were obtained at 11.9 kV/cm of driving field amplitude. Error bars represent measurement error. These errors alongside the details of the annealing procedure of the coarse-grained BFO sample (average grain size 16 μm), its microstructure, domain structure and local electrical conductivity are reported in Ref. [79]. The two insets show the time-domain driving field E signal (black sinusoidal curves) with overlaid mechanical displacement ΔL signals (blue or green sinusoidal curves) together with the corresponding ΔL-E piezoelectric hysteresis loops for the case of (b) positive (10 Hz; blue data) and (c) negative (1 Hz; green data) piezoelectric phase angle. The bigger arrows in panels (b) and (c) indicate the lagging and leading ΔL signal with respect to the driving field signal for the case of positive and negative phase angle, respectively. The corresponding counter-clockwise and clock-wise rotational sense of the hysteresis, which is set by the lagging and leading output signals, respectively, is also noted on the respective loops. Numbered points on the time-domain plots and hysteresis are drawn to help identifying the rotational sense of the hysteresis.*

rise to a d33 maximum at 1.5 Hz (**Figure 3a**, black points). These two d33-vsfrequency behaviors are accompanied by a broad +tanδ<sup>p</sup> maximum at 10 Hz and a rather narrow –tanδ<sup>p</sup> minimum at 0.4 Hz, respectively (**Figure 3a**, red points). The overall behavior can be thus simplified to a sequence of retardation process (d33 increasing with decreasing frequency accompanied by +tanδ<sup>p</sup> peak) and relaxation process (d33 decreasing with decreasing frequency accompanied by –tanδ<sup>p</sup> peak) [4, 46]. As it will be shown in the next section, this is consistent with the distinct features of M-W piezoelectric effect, which can be easily predicted by simple analytical modeling.

As it has been done in the case described here, the sign of the piezoelectric phase is very often determined by measurements performed using lock-in technique, which is capable of extracting the amplitude and phase of individual harmonic responses to external sinusoidal excitations [29]. For less experienced, this may be sometimes nontrivial as several instruments may reverse (by 180°) or somewhat affect the phase of the output signal. While the sign of tanδ<sup>p</sup> can be checked by, e.g., measurements of a sample with known response, such as, e.g., a donor-doped soft PZT where tanδ<sup>p</sup> should be positive (and, ideally, related to Eqs. (1) and (2)), it is useful to acquire the total signal containing all information by conventional oscilloscope imaging. Two examples of such acquired signals displayed either in time domain or as piezoelectric

hysteresis loop (mechanical displacement ΔL versus electric field E plots) are shown in **Figure 3b** and **c**. The case in **Figure 3b** is shown for 10 Hz where a positive tanδ<sup>p</sup> was measured using the lock-in method. As expected, the output displacement ΔL signal lags behind the input driving field E signal (see arrow in **Figure 3b**), corresponding to counter-clockwise rotational sense of the piezoelectric hysteresis. In terms of hysteresis, this is a common situation observed during, e.g., conventional measurements of ferroelectric P-E hysteresis loop at switching fields (to give a popular example). The case that is less common is when tanδ<sup>p</sup> is negative as shown in **Figure 3c**. Here, instead of *lagging*, the output ΔL signal *leads* the input E signal, effectively resulting in a clock-wise rotation of the piezoelectric hysteresis.

Very often (and not to be blamed) the negative piezoelectric phase is misinterpreted because it gives a wrong impression that it violates the basic law of energy conservation. This is also the reason why is so often interpreted as a measurement artifact. An example is the clockwise hysteresis measured in the response of ferroelectric field-effect transistor thin-film structures, which arise due to charge injection during measurements [86, 87].

It is not within the scope of this contribution to provide a deep and detailed physical analysis of the negative piezoelectric phase, neither is such analysis within the main expertise of the author of this chapter. The reader is strongly advised to follow the discussion provided in the chapter on hysteresis by Damjanovic [4]. Nevertheless, one can envision a very simple reasoning based on the classical power dissipation principles in dielectrics (found in general textbooks, such as [88]) where the power loss is defined by the positive dielectric tanδ (considering that all other parameters in the equation for the dissipated power density, i.e., electric-field amplitude, frequency and real part of the dielectric permittivity, are positive by definition). Similarly, the area of the charge density (D) – electric field (E) hysteresis represents the energy loss of a dielectric during an electric field cycle per unit volume of the material (units of D-E hysteresis is J m�<sup>3</sup> ) [89]. In this analogy, one could understand a negative piezoelectric phase angle corresponding to energy gain that is represented by the piezoelectric hysteresis area. The simplest and perhaps strongest argument against this claim is that, unlike the dielectric D-E hysteresis area, the piezoelectric hysteresis area (either converse x-E or direct D-Π) does not have units of energy density. Therefore, the hysteresis shown in **Figure 3c**, where the phase is negative, does not *directly* represent an energy gain because the hysteresis area does not reflect energy density. This problem has been extensively elaborated by Holland [90]. The rigorous mathematical treatment therein shows that the necessary restriction of positive total power loss, which is represented by the sum of dielectric, elastic and mix piezoelectric components (proportional to respective imaginary coefficients), results into dielectric and elastic loss terms being always positive and bigger than the piezoelectric term. The latter was shown to not be restricted in its sign. Therefore, taking the longitudinal piezoelectric response of poled ferroelectric ceramics as an example, the imaginary piezoelectric coefficient *d33"* is permitted to be either positive or negative; for a positive real longitudinal coefficient *d33'*, as is the case of ferroelectric ceramics, the piezoelectric tanδ<sup>p</sup> (defined as *tan δ<sup>p</sup>* ¼ *d*00 33 *d*0 33 ) can thus be either positive or negative. While not restricted in its sign, the piezoelectric term in the total power dissipation function is, however, restricted in magnitude. This essentially means that a negative piezoelectric phase angle measured in poled ceramics indicates a partial reduction of the total power loss. To avoid confusion, it should be noticed that in the power dissipation equation, reported in the paper by
