*2.3.5.1. Development of an internal bias*

It is well known that domain switching occurs through a nucleation-growth process of reversed domains driven by the *local electric field Eloc(r, t)*, which is given by the sum of the following fields: i) the external applied electric field *Eapp*; ii) the *depolarization field Edep(r, t)* due to the polarization changes produced by the formation of reversed nuclei (nucleation) at the location r and time t during the application of the external field *Eapp*; iii) the sum of the *screening fields*, which can be divided into: a) *external screening field Ees(r, t)* produced by free charges on electrodes, and b) *bulk screening field Ebs(r, t)* caused by the rearrangement of charge carriers in a ferroelectric [29]:

$$E\_{\rm loc}\left(r,t\right) = E\_{app} - \left\{E\_{dep}\left(r,t\right) - \left[E\_{es}\left(r,t\right) + E\_{bs}\left(r,t\right)\right]\right\} \tag{3}$$

In an ideal case of a perfectly insulating poled ferroelectric placed between two conductive plates of a capacitor, the polarization bound charges in the ferroelectric will cause an accu‐ mulation of free electronic charges on the electrodes in the region nearby the ferroelectric to counterbalance the polarization charges, thereby leading to the presence of *a depolarizing field* inside the ferroelectric. By short circuiting the capacitor plates, the compensation charges will flow within the circuit in such a way that the depolarizing field disappears (*the external screening*). In real ferroelectric capacitors, however, the electrodes are not perfect conductors and the ferroelectric would present the so called *dielectric gaps* in the interior and close to the interface with the electrodes, where the spontaneous polarization is significantly suppressed or absent. [30] The presence of dielectric gaps determines a separation between the compen‐ sation charges and the polarization bound charges in the ferroelectric. The former become trapped at the dielectric gap-ferroelectric interface forming a space charge layer, which, together with the polarization bound charges in the ferroelectric, generates a *depolarizing* *field*, which can only be partially compensated by external screening in the short circuit condition. The time constant of the external screening process is determined by the parameters of the external circuit and it is typically in the order of micro/nano seconds. [30] The unscreened part of the depolarizing field in the interior of the ferroelectric, *a residual depolarizing field Eres*, is antiparallel to the polarization, and triggers the so called *bulk screening process*, which involves a rearrangement of the charge carriers inside the ferroelectric to reduce the residual depolarizing field. The bulk screening process can occur through: i) redistribution of charge carriers, ii) alignment of dipolar defects, or iii) irradiation-induced charge injection. The time constant of the bulk screening process is typically several orders of magnitude higher than that of the external screening and it often exceeds the switching time of the ferroelectric polariza‐ tion. Therefore, the separated and trapped screening charges at the gap/ferroelectric interface can cause an internal bias field which points in the direction of the polarization.

The non-ergodic relaxor BNT-BT has shown domain fragmentation during fatigue [28], while the ergodic compositions exhibit significantly higher fatigue resistance [27]. The current understanding is that the domain wall pinning effects become less significant in ergodic relaxor phases.[23] In addition, it can be considered that the ergodic relaxors return to a weakly polar state with low remanent polarization and low remanent strain during electric field unloading. This yields smoother variations of polarization and strain during cycling, which could probably be one of the reasons of the less pronounced fatigue effects. However, further studies are needed to better elucidate the mechanisms of the increased fatigue resistance in lead-free

Macroscopic fatigue-like effects such as asymmetric hysteresis loops are often similar in different material systems, possibly due to a common origin represented by the presence of an internal bias field. However, the microscopic origin of the internal bias could be different

It is well known that domain switching occurs through a nucleation-growth process of reversed domains driven by the *local electric field Eloc(r, t)*, which is given by the sum of the following fields: i) the external applied electric field *Eapp*; ii) the *depolarization field Edep(r, t)* due to the polarization changes produced by the formation of reversed nuclei (nucleation) at the location r and time t during the application of the external field *Eapp*; iii) the sum of the *screening fields*, which can be divided into: a) *external screening field Ees(r, t)* produced by free charges on electrodes, and b) *bulk screening field Ebs(r, t)* caused by the rearrangement of charge carriers in

*E rt E E rt E rt E rt loc* ( , ) =- - + *app dep* { (,) (,) (,) é ù

In an ideal case of a perfectly insulating poled ferroelectric placed between two conductive plates of a capacitor, the polarization bound charges in the ferroelectric will cause an accu‐ mulation of free electronic charges on the electrodes in the region nearby the ferroelectric to counterbalance the polarization charges, thereby leading to the presence of *a depolarizing field* inside the ferroelectric. By short circuiting the capacitor plates, the compensation charges will flow within the circuit in such a way that the depolarizing field disappears (*the external screening*). In real ferroelectric capacitors, however, the electrodes are not perfect conductors and the ferroelectric would present the so called *dielectric gaps* in the interior and close to the interface with the electrodes, where the spontaneous polarization is significantly suppressed or absent. [30] The presence of dielectric gaps determines a separation between the compen‐ sation charges and the polarization bound charges in the ferroelectric. The former become trapped at the dielectric gap-ferroelectric interface forming a space charge layer, which, together with the polarization bound charges in the ferroelectric, generates a *depolarizing*

ë û *es bs* } (3)

in different systems and strongly dependent on the type of electrical loading.

ergodic relaxors.

a ferroelectric [29]:

*2.3.5. Microscopic mechanisms of fatigue-like effects*

214 Ferroelectric Materials – Synthesis and Characterization

*2.3.5.1. Development of an internal bias*

In uniaxial ferroelectric ceramics where the dipolar defects are constrained, the main physical mechanism responsible for aging and fatigue phenomena is attributed to the compensation of the residual depolarizing field through the redistribution of the screening charges. The idea employed in this scenario assumes that free charges present in the material migrate to minimize the depolarizing field with the consequent development of an internal bias field that hampers domain wall movement and generates biased polarization and strain states (Fig. 5). Balke *et al.* [30] proposed that the depolarizing field *Edep* surrounding a given grain in samples fatigued under an applied field *Eapp* can be estimated as:

$$E\_{\rm dep} = -\alpha\_{\rm avg,lo} \frac{\Delta P}{\mathcal{E}\_0 \mathcal{E}\_{33}} \tag{4}$$

where *ΔP* is the variation of polarization during the increase of the applied field from *E = 0* to *E = Eapp* and *ε*33 is the permittivity at *E = Eapp*. The factor *αangle*, which can assume only values between 0.15 and 0.5, takes into account the partial compensation of the depolarization field based on the orientation of the polarization in neighboring grains. This model was able to predict the range of internal bias fields observed in DC poled samples of Pb0.99[Zr0.45Ti0.47(Ni0.33Sb0.67)0.08]O3 at room temperature.[11, 30] Additionally, this mechanism was invoked to explain the fatigue-like effects after unipolar cycling in Pb0.99[Zr0.45Ti0.47(Ni0.33Sb0.67)0.08]O3 at room temperature [11] and it can be also applied to rationalize the biasing effects observed in DC poled hard piezoceramics, as shown in Fig. 4c.

In the case of bipolar fatigue, the bulk screening process under an alternating electric field leads to an inhomogeneous internal field distribution, which yields the development of the so called *frozen domains*. These get locked and do not switch after a certain number of electric field cycles, giving rise to heterogeneous fatigue effects. The value of *Ebs(r, t*) in the Eq. 3 is deter‐ mined by the sample history, including sintering/annealing, the subsequent deposition of electrodes, electrical cycles characteristics (frequency and time exposure) and time intervals between cycles.[31] These factors have been taken into account in the *kinetic imprint approach*, which was successfully employed in describing the hysteresis loops after fatigue in different materials. [32]

**Figure 5.** Schematic of the formation of an internal bias after poling.
