Thermal stability of the electromechanical properties in acceptor-doped and composite-hardened (Na_1/2Bi_1/2)TiO₃–BaTiO₃ ferroelectrics

There is an ever-increasing demand for hard-type lead-free ferroelectrics^1–3 due to their applicability in piezoelectric transducers and ultrasonic motors. This demand is not only triggered by the toxicity of lead and the related legislations⁴ but particularly by lead-free compositions outperforming Pb(Zr,Ti)O₃ (PZT) based ceramics in specific high-power piezoelectric applications.^5–8 One of the most promising compositions is the relaxor (1–x)Na_1/2Bi_1/2TiO₃–xBaTiO₃ (NBT–100xBT),⁹ which transforms into a ferroelectric upon the application of a large electric field, E.^10,11 A morphotropic phase boundary (MPB) at 6–7 mol. % BaTiO₃⁹ is associated with the reasonably high small-signal piezoelectric parameters (piezoelectric coefficient d₃₃ = 120–150 pm/V and coupling coefficient k₃₃= 0.55^9–11). Moreover, the material exhibits very stable electromechanical properties over a broad vibration velocity range.^5,6 This stability is key to increase the output power in next-generation high-power devices. However, the overall low electromechanical quality factor, $Qij$⁠, of about 150;¹² a depolarization temperature, $Td$⁠, below 100 °C;¹³ and the complicated depolarization mechanisms^14–16 are among the main deficits of NBT–6BT as well as of related NBT–20KBT.¹⁶ 

Thermal depolarization is described as the thermally induced disappearance of macroscopic polarization that ultimately results in electromechanical coefficients being reduced to zero. The depolarization mechanisms are correlated to the complex electric field–temperature (E–T) phase diagram of NBT–6BT due to its relaxor ferroelectric nature. Mechanistic processes are presented in the simplified schematic (Fig. 1), which resembles the present understanding of the complex behavior of NBT–BT, based on the experimental data and the model of polar nanoregions (PNRs). Below the freezing temperature,¹⁷ the polarization direction fluctuation is thermally inhibited [Fig. 1(a) (I)]. In this non-ergodic relaxor state, the material can be polarized by the application of a large electric field, inducing an irreversible long-range ferroelectric order [Fig. 1(a) (II)], i.e., in the case of NBT–6BT, a rhombohedral/tetragonal phase mixture.^10,11 Heating the material above the ferroelectric-to-relaxor transition temperature, $TF−R$⁠, reinstates the polarization fluctuation, reverting the material to the ergodic state [Fig. 1(a) (VI)].

Thus, $TF−R$ sets the natural upper limit to the depolarization boundary. However, several other processes can lead to depolarization as well, including detexturization of the domain polarization directions in a polydomain material and a ferroelectric-to-ferroelectric phase transition.¹⁸ Studies on NBT-based compositions, including NBT–6BT, proposed a two-stage depolarization process^14–16 [Fig. 1(a) (III) and (IV)]. Temperature-dependent piezoelectric coefficient $(d33)$ and thermally stimulated depolarization current (TSDC) measurements $(Td)$ in contrast to temperature-dependent relative permittivity, $εr$⁠, measurements $(TF−R)$ revealed about 8 °C lower $Td$ values as compared to $TF−R$ in NBT–6BT.^14,19 It was suggested that the depolarization sequence starts with detexturization of the domain polarization, like in classical ferroelectrics. Thus, the macroscopic polarization vanishes prior to the ferroelectric-to-relaxor transition and the recovery onset of the ergodic relaxor state. This finding was supported by x-ray diffraction (XRD),¹⁵ which did not give unambiguous evidence for the phase transition around $Td$⁠.

However, other studies on NBT–6BT did not observe a clear shift between $Td$ and $TF−R$⁠,^13,20 despite utilizing the same measurement methods. Also, a temperature-induced increase of the non-polar pseudocubic phase at the expense of the ferroelectric phase around T_d (=T_F−R) was noted,^21–23 indicating a second-order phase transition [Fig. 1(a) (V)] rather than a sharp displacive first-order transition like in classical ferroelectrics or Pb-based relaxors.²⁴ 

Orientation-dependent single-crystal measurements of different NBT–100xBT compositions revealed that both scenarios are possible because $Td$ and $TF−R$ are not necessarily linked.²⁵ $TF−R$ depends on the thermal stability of the local coherence of neighboring dipole moments, which breaks up upon heating above $TF−R$⁠. $Td$ depends on the thermal stability of the long-range coherence of polarization directions in the neighboring domains, i.e., on the domain configuration. It was observed that NBT–100xBT crystals, particularly MPB compositions with a rhombohedral/tetragonal phase mixture, exhibited orientation and domain configuration dependent $Td$ and $TF−R$ values, where $Td$ was lower, higher, or equal to $TF−R$⁠. This indicates that both, a direct transition into the ergodic relaxor state and prior detexturization of domain polarization, can induce depolarization. Moreover, it was demonstrated that also ferroelectric-to-non-ergodic relaxor transition appears upon heating, which means that the ferroelectric domains break up into PNRs. Thereby, the polarization directions of the PNRs remain aligned in the original poling direction. Additionally, it was pointed out that the ferroelectric rhombohedral and tetragonal phases do not necessarily exhibit equal $TF−R$ values $(TF−Rtetr.≠TF−Rrhom.)$⁠, which is usually masked by a relatively broad anomaly in the dielectric permittivity but appears rather distinct in TSDC measurements.^19,25,26

Several approaches have been utilized to increase the $Td$ or the $Qij$ of NBT–100xBT-based piezoceramics, including chemical doping,^20,21,27 quenching,^28–30 grain size engineering,³¹ off-stoichiometry,³² and ceramic–ceramic composite formation.^12,13,33 A substantial and simultaneous increase of $Td$ and $Qij$ was achieved by Zn²⁺ doping²⁰ and composite formation with the ZnO secondary phase.^12,13

The $Td$ increase by Zn²⁺ doping is referred to the enhancement of the rhombohedral and tetragonal distortion and the large polarizability of Zn²⁺.²⁰ Both stabilize the long-range ferroelectric order and increase the thermal stability.²¹ Hardening by Zn²⁺ doping is analogous to the acceptor-doping methodology used in PZT.^34,35 The Zn²⁺ substitution at the perovskite B-site is compensated by oxygen vacancy formation to maintain charge neutrality. This generates lattice defects or defect complexes that hinder domain wall motion and reduces the losses [Fig. 1(b) (II)].

The recently introduced second-phase hardening concept in 0–3 ceramic–ceramic composites^12,13,33 is less well understood [Fig. 1(b) (III)]. Current understanding relates the increased thermal stability and the induced ferroelectric hardening to mechanical mismatch stresses arising from the strain incompatibility between the matrix and the inclusions. The mismatch stresses evolve after cooling the composite from the sintering temperature due to thermal expansion differences between the matrix and the inclusions³⁶ as well as upon application of an electric field to the ferroelectrically switching NBT–6BT matrix phase and the non-ferroelectric ZnO inclusions.¹² However, recent TEM studies indicated Zn²⁺ diffusion into the NBT–6BT lattice, suggesting at least an additional doping contribution in the composites, which aggravates the deconvolution of the mechanisms,³⁷ similarly as discussed for Pb(Zr,Ti)O₃–ZrO₂ composites.³⁸ The composite approach is considered as a generic hardening concept because it does not require a chemical modification but relies on the inclusion of a tailored second phase. The concept is extendable to other ferroelectric systems and has been demonstrated or at least indicated on PZT-based,³⁸ NBT-KBT-based,³⁹ and NBT-based^12,33 compositions. However, its temperature dependence and depolarization processes are not well understood yet.

In this work, the depolarization processes and hardening mechanisms of Zn²⁺-doped NBT–6BT and NBT–6BT/ZnO composites are compared with undoped NBT–6BT and acceptor-doped PZT in a broad temperature range. After a description of microstructures (Sec. III A), the thermal depolarization process is studied (Sec. III B) and the observed extension of the depolarization temperature range is discussed, which is associated with decoupling of the longitudinal and transverse piezoelectric coefficients. This is followed by the investigation of the hardening mechanisms through the temperature dependence of the small-signal electromechanical parameters (Sec. III C) and their degradation (Sec. III D) as well as the large-signal properties (Sec. III E). The underlying mechanistic differences between the investigated materials are elucidated by the temperature dependence of polarization harmonics (Sec. III F).

Samples of five 0.94Na_1/2Bi_1/2TiO₃–0.06BaTiO₃ (NBT–6BT) compositions, including an undoped reference, were prepared by the conventional solid-state reaction route. Two (Na_1/2Bi_1/2)_0.94Ba_0.06Ti_100−yZn_yO₃ compositions were modified by B-site substitution with y mol. % Zn (y = 0.5 and 1; NBT–6BT–yZn), while two others were modified by composite formation with z % mole-ratio of nm-sized ZnO (z = 10 and 40; NBT–6BT:zZnO). Detailed processing conditions for undoped, doped,¹² and composite¹³ compositions were reported earlier. The reference hard PZT samples were cut from a sintered block of a commercial material (SONOX® P4, CeramTec GmbH).

Samples were cut and ground into bars of 20 × 3 × 1 mm³ (small-signal measurements) or disks of 6 mm diameter and 0.6 mm thickness (large-signal measurements) and subsequently annealed at 400 °C with a heating rate of 5 K/min. Electrodes were fabricated by Au-sputtering on the two faces with the largest surface area. All samples were poled with 6 kV/mm for 20 min at 20 °C and aged 24 h before further measurements.

The microstructure was imaged by scanning electron microscopy (SEM; XL30 FEG, Philips Corporation) using a backscattered electron detector. The samples were polished to a 0.25 μm finish with diamond paste and thermally etched at 1000 °C for 10 min. Temperature-dependent permittivity measurements were carried out at 0.1, 1, and 10 kHz with 10 mV/mm AC amplitude and a heating rate of 2 K/min using an impedance analyzer (Alpha-Analyzer, Novocotrol Technologies GmbH). The poling process was evaluated by measuring $d33$ with a Berlincourt meter (PiezoMeter System PM 300, Piezotest Pte Ltd.) at 110 Hz. The inverse temperature-dependent piezoelectric coefficients, $d33$ and $d31$⁠, were measured in situ using a laser vibrometer (OFV-505 sensor and VDD-E-600 front-end, Polytec GmbH) with a 2 K/min heating rate at 1 kHz and 10 V/mm AC amplitude. The bar shape and dimension ratios were chosen to excite the transverse (31) vibration mode.⁴⁰ Small-signal electromechanical coefficients (⁠$Q31R$⁠, $k31$⁠, $d31$⁠, and $s11E$⁠) were determined by resonance impedance spectroscopy and the 3 dB calculation method, according to Ref. 40, using an impedance analyzer (Alpha-Analyzer, Novocotrol Technologies GmbH). Temperature-dependent polarization (P–E) and strain (S–E) hysteresis loops were recorded with a TF Analyzer 2000 (aixACCT Systems GmbH). A triangular field with 1 Hz frequency and a maximum amplitude of 6 kV/mm was utilized. Polarization harmonics measurements were carried out using a customized shunt-resistor circuit.⁴¹ Excitation voltage on the sample, as well as the voltage drop on the shunt-resistor, along with the current–voltage phase angle, were generated and determined by a lock-in amplifier (SR830 DSP, Stanford Research). The excitation voltage amplitude was enhanced by a voltage amplifier (PZD700M/S, Trek Inc.). A sinusoidal voltage signal with 1 kHz frequency was utilized; the samples were preconditioned and measured with a descending voltage amplitude. The samples were heated at 2 K/min and thermally stabilized for 30 min at each measurement temperature.

All electromechanical measurements were carried out on two samples of each composition and each sample was measured three times. The presented results are, thus, averaged over six measurements; uncertainties were calculated by the standard deviation according to the Gaussian uncertainty propagation. The samples were reloaded several times at room temperature to exclude any influence from positioning and clamping.

The microstructures and grain size distributions of representative NBT–6BT, NBT–6BT:10ZnO, and NBT–6BT–1Zn samples are displayed in Fig. 2 (for other compositions, see Fig. S1 in the supplementary material). No secondary phases can be observed in the undoped and doped samples, while the grain size distribution becomes much broader after doping. The NBT–6BT:10ZnO microstructure consists of ZnO inclusions [red arrows in Fig. 2(b)] embedded in the NBT–6BT matrix, while the grain size distribution remains narrow. In addition, a small amount of another secondary phase can be observed (blue arrow), which was attributed to a TiZn₂O₄ spinel phase.¹³ Detailed phase analysis was published previously.^13,33

The average grain size of the NBT–6BT phase increases by a factor of 2 in composites and a factor of 4–8 in Zn²⁺-doped samples, compared to the undoped NBT–6BT (Figs. 2 and S1 in the supplementary material). It was reported that a large increase in grain size can lead to enhanced $Td$ and lower $d33$ values.³¹ However, such a trend cannot be observed when all compositions are taken into account. Table I shows the determined $Td$ and $d33$ values, which will be discussed in Secs. III B–III D. In fact, although the average grain size increases by a factor of 2 in NBT–6BT–1Zn, as compared to NBT–6BT–0.5Zn, the changes in $Td$ and $d33$ are only minor, below 6%. Therefore, it is unlikely that the observed hardening and increase of $Q31R$ (Table I) can be attributed to the grain size variation because severe changes of the ferroelectric properties associated with modified domain configurations would only be expected in the sub-μm grain size range.⁴² Thus, the grain size dependence on hardening will be neglected in the subsequent analysis.

The thermally induced transition from the poled ferroelectric state into the relaxor state, $TF−R$⁠, was determined by measuring the inflection point of the temperature-dependent real part of the relative permittivity, $ε′r$⁠, and the dielectric loss, $tanδ$⁠, at the thermal anomaly [Figs. 3(a)–3(c); for other compositions, see Fig. S2 in the supplementary material]. The $TF−R$ increases from 98 °C in NBT–6BT to 131 °C in NBT–6BT:10ZnO and 115 °C in NBT–6BT–1Zn (Table I), which is in agreement with previous reports.^13,21,33 Furthermore, composites and Zn²⁺-doped samples feature an additional low-temperature anomaly starting at 86 °C. The smaller low-temperature anomaly is best seen in the inverse $1/ε′r(T)$ representation and highlighted by the shaded region [Fig. 3(d)]; the onset was determined from the first derivative of $1/ε′r(T)$ (Fig. S3 in the supplementary material).

Thermal depolarization is described as the thermally induced disappearance of macroscopic polarization that ultimately results in electromechanical coefficients being reduced to zero. However, the polarization decrease can occur over a wide temperature range, particularly in NBT-based materials; therefore, it was suggested to set T_d at the temperature of the strongest polarization loss rate,¹³ although a residual polarization might still exist even at higher temperature. $Td$ was, thus, determined from the inflection point of the inverse piezoelectric coefficient $d33$ measured in situ during heating [Fig. 4(a) and Table I]. NBT–6BT exhibits an initial increase, followed by a sharp drop of $d33$ at 99 °C, while NBT–6BT:10ZnO and NBT–6BT–1Zn display a smaller increase of $d33$ and a delayed depolarization at 133 and 120 °C, respectively. Interestingly, the $d31$ coefficients feature a significantly different temperature dependence [Fig. 4(b)]. $d31$ increases steadily in all compositions in the low-temperature regime to about 85 °C. At higher temperatures, $d31$ of undoped NBT–6BT starts to decline before it drops to almost zero at 99 °C. However, in the two modified compositions, $d31$ decreases gradually above 85 °C, ending in a final drop at the previously determined $Td$ values. These temperature regimes coincide with the anomaly onset determined from the permittivity measurements (Fig. 3).

The differences between $Td$ and $TF−R$ values are within the small error ranges (Table I) and can, thus, be considered equal, which is in good agreement with previous reports.^13,20,23 This indicates that the depolarization process is predominantly dictated by and directly terminates in the ferroelectric-to-relaxor transition [Fig. 1(a) (V)], which has also been proven in recent structural studies of undoped NBT–6BT.²³ The absence of a distinct separation of $Td$ and $TF−R$⁠, as observed at least for undoped NBT–6BT in other studies,^14,15 might be induced by a more stable domain configuration, related, for example, to a larger grain size [as was observed in BaTiO₃⁴³ and Pb(Zr,Ti)O₃⁴⁴].

The higher depolarization temperatures demonstrate the efficient stabilization of the electromechanical response by Zn²⁺ doping and even more by ZnO inclusions. However, the appearance of the anomaly at 86 °C suggests that although both types of modifications modify the T_d (= T_F−R), they do not shift but rather extend the temperature range of thermal depolarization. This means that the depolarization onset remains at 86 °C, nearly unchanged as compared to undoped NBT–6BT (Fig. 3). This specific temperature is consistent with the temperature-dependent XRD studies of undoped and Zn²⁺-doped NBT–6BT.^15,21,23 These studies identified this temperature range as the onset of a phase transition into the ergodic relaxor state by the appearance of a pseudocubic phase and demonstrated that the transition is rather a continuous process over a range of several °C. The $ε′r$ anomaly, the increasing $εr′$ and $tanδ$ values, and the increasing frequency dispersion starting at 86 °C (Fig. 3) suggest that depolarization is induced by gradual nucleation and growth of the non-polar ergodic relaxor phase (accelerated at higher temperatures), while the domain polarization texturization remains mostly unchanged [Fig. 1(a) (V)]. Considering the previously reported different $TF−R$ values of the rhombohedral and tetragonal phases,^19,23,25 it is likely that the modifications by Zn²⁺ doping and ZnO inclusions result in a shift of the $TF−R$ of the tetragonal phase, while the rhombohedral phase remains unaffected. This hypothesis is based on experimental findings, which demonstrated that the $TF−R$ of the tetragonal phase can be strongly perturbed and shifted, for example, by an external electric field,¹⁹ the induced domain configuration,²⁵ or the BT content,²³ while the $TF−R$ of the rhombohedral phase remains nearly unchanged.

Interestingly, the depolarization process is associated with decoupling of the $d33$ and $d31$ coefficients (Fig. 4). The origin of this behavior is not entirely understood, and, to the authors' knowledge, such behavior has not been previously reported for relaxor ferroelectrics. Also, decoupling is not observed in the $Td$ vicinity of classical ferroelectrics²¹ nor is it predicted in the Landau–Ginzburg–Devonshire theory framework.⁴⁵ However, comparable observations are often reported in piezoelectric two-phase materials,^46,47 where a piezoelectric powder is embedded in a non-piezoelectric matrix and the decoupling increases with a decreasing ratio of the piezoelectric phase. Following the aforementioned picture of the depolarization mechanism, the material structure beyond the depolarization onset temperature (≈86 °C) can be interpreted as a two-phase structure with the residual ferroelectric phase embedded in the recovered non-polar pseudocubic ergodic relaxor phase [Fig. 1(a) (V)]. A similar proposition has been made before.²⁵ Thereafter, we hypothesize that the decrease of the $d31$ coefficient in the depolarization range is induced by the two-phase structure formation, while the $d33$ simultaneously increases continuously until $Td$⁠. From the application perspective, the decoupling opens up interesting advantages for the usage of these materials in directional ultrasonic transducers, demonstrated by an enhanced hydrostatic piezoelectric coefficient, $dh$⁠, in a broadened temperature range [Fig. 4(c)].

Figure 5 provides the temperature dependence of the small-signal coefficients $Q31R$⁠, $k31$⁠, $d31$⁠, and $s11E$ measured in situ between −30 and 140 °C using resonance impedance spectroscopy. A comparison of all NBT–BT compositions to a prototype hard PZT is given in Fig. S4 in the supplementary material. The depolarization is marked by the disappearance of (anti)resonance peaks from the impedance spectrum. These $Td$ values are in good agreement with the values from off-resonance measurements, albeit slightly lower (Table I; note the measurement step size of 5 °C). Undoped NBT–6BT exhibits a steady decrease of $Q31R$ with increasing temperature up to depolarization [Fig. 5(a)]. On the other hand, all modified NBT–6BTs reveal an increase and a subsequent decrease of $Q31R$ with a maximum at about 30 °C [Fig. 5(a)]. This maximum is particularly pronounced in NBT–6BT–1Zn, while NBT–6BT:10ZnO demonstrates the most invariant temperature behavior with a broad maximum. Interestingly, the behavior of NBT–6BT–1Zn resembles the temperature dependence of hard PZT [Fig. S4(a) in the supplementary material], which supports the acceptor-doping hardening mechanism [Fig. 1(b) (II)].

The temperature dependences of $k31$ and $d31$ are similar in all NBT–6BT compositions [Figs. 5(b) and 5(c)]. $k31$ and $d31$ feature a continuous increase until about 80 °C (depolarization onset) and a subsequent degradation until reaching the respective $Td$⁠. Zn²⁺-doping and ZnO composite formation significantly extend the depolarization range. However, although $Td$ increases, the onset of the depolarization is not shifted, as previously discussed in Sec. III B.

The $s11E$ increases continuously with increasing temperature and does not resemble the temperature dependence of other coefficients. The curve exhibits no anomaly at the depolarization onset [Fig. 5(d)], and only a small decrease is observed in the vicinity of $Td$⁠.

The small-signal electromechanical coefficients of all NBT–6BT samples exhibit very similar temperature dependencies, except for $Q31R$⁠. Especially, the $Q31R$ behavior of the Zn²⁺-doped NBT–6BTs and hard PZT, i.e., the increase and subsequent decrease with increasing temperature, is not fully understood yet. Note that there is no evidence for a phase transition in this range and the behavior does not follow the trend of $s11E$⁠. It appears that at least two contributions influence the observed temperature dependence of $Q31R$⁠, revealed by the slope change from positive to negative at 30 °C. We hypothesize that the increase in $Q31R$ at low temperatures (positive slope) results from the temperature-dependent inclination of the MPBs in the compositional phase diagrams induced by Zn²⁺ doping. With increasing temperature, the compositions shift away from the MPB, which usually results in ferroelectric hardening due to increased lattice distortion and consequently in an increase in $Q31R$ values.^5,48,49 However, the increasing temperature also thermally enhances the domain wall mobility and reduces the pinning strength of oxygen vacancies $VO⋅⋅$ and/or $VO⋅⋅−Zn′′Ti$ defect complexes due to thermally activated $VO⋅⋅$ migration.⁵⁰ The enhanced mobility of the domain walls leads to a decrease of the $Q31R$ values^51–53 and an increase of $k31$ and $d31$⁠.

On the other hand, the domain wall pinning mechanism in NBT–6BT:xZnO composites relies on mechanical stress induced by the strain incompatibility between the switching NBT–6BT and non-switching ZnO phase [Fig. 1(b) (III)] and is, thus, only marginally altered in the investigated temperature range. The slight $Q31R$ change at about 30 °C can be explained by the minor Zn²⁺ diffusion from ZnO inclusions into the NBT–6BT lattice, inducing the same effect as in the doped compositions. The relatively stable temperature behavior of composites confirms that the dominant hardening mechanisms in doped samples and composites are fundamentally different. It is also remarkable that the depolarization onset impacts the temperature dependence of $k31$ and $d31$ but not of $Q31R$ and $s11E$⁠. Please note that all changes appearing above the depolarization onset are irreversible and the values do not recover upon cooling (Fig. S5 in the supplementary material).

To further probe the degradation behavior of the NBT–6BT compositions, time-dependent measurements of the electromechanical properties at constant temperature were performed (Fig. 6). The temperature was chosen to be in the vicinity of $Td$ of the respective sample (90 °C for NBT–6BT and 110 °C for both modified NBT–6BT compositions), where a substantial degradation of the electromechanical coefficients with increasing temperature was observed (Figs. 4 and 5). $Q31R$⁠, $k31$⁠, $d31$⁠, and $s11E$ were tracked continuously for 12 h under small-signal excitation. NBT–6BT exhibited no change of $Q31R$⁠, while $k31$⁠, $d31$⁠, and $s11E$ decrease by less than 10% and stabilized after 4 h. NBT–6BT–1Zn exhibited a decrease of $Q31R$ by 8% and stabilized after 5 h. $k31$⁠, $d31$⁠, and $s11E$ did not show a degradation. Interestingly, no changes were observed for NBT–6BT:10ZnO during the entire time. The very small time-dependencies indicate that these materials' changes, leading to the strong temperature dependence of the electromechanical properties (Figs. 4 and 5), are fast and thermally stable processes.

To evaluate the temperature-dependent domain wall pinning strength of the different hardening mechanisms, the P–E and S–E hysteretic loops of NBT–6BT, NBT–6BT:10ZnO, and NBT–6BT–1Zn in the poled and aged state are depicted in Fig. 7 (loops at elevated temperatures are given in Figs. S6 and S7 in the supplementary material). The loops of NBT–6BT:10ZnO and NBT–6BT–1Zn reveal a distinct asymmetry and a shift along the abscissa, which both decrease at higher temperatures. The shift can be quantified by the internal bias field, $Eib$⁠,³⁴ which is taken as the half-difference between the positive and negative coercive fields, $Ec$⁠, determined from the strain minima [Figs. 7(c) and 7(d)]. At 20 °C, the average $Ec$ values of NBT–6BT:10ZnO and NBT–6BT–1Zn are 0.12 and 0.74 kV/mm larger as compared to that of NBT–6BT, respectively, while $Eib$ exhibits a threefold increase in NBT–6BT:10ZnO and a fourfold increase in NBT–6BT–1Zn, as compared to that in NBT–6BT. Both characteristics clearly indicate ferroelectric hardening of the modified compositions, which is also reflected by the larger $Q31R$ values (Table I).

The average electric coercive fields strongly decrease at higher temperatures in all materials due to the thermal activation of the polarization switching process and the reduced lattice distortion. However, the temperature dependence of $Eib$ reveals two distinct temperature regimes: a decrease from 20 to 70 °C and an increase with an irregular trend above 80 °C [Fig. 7(d)]. The boundary between both regimes coincides with the onset of thermal depolarization (Figs. 3 and 4). Therefore, the hysteresis loops represent the convoluted polarization and strain responses of the ferroelectric and the ergodic relaxor phases with a temperature-dependent phase ratio. Evidence for the mixed-phase state above 80 °C are the evolving slim polarization loops, sprout-shaped strain loops with small negative strain, and the change of the remanent state before and after applying the electric field (Figs. S6 and S7 in the supplementary material). Note that the P–E loops measured at the highest temperature of each composition show an additional minor contribution from electrical conductivity, which is indicated by the appearance of round hysteresis tips and arises from an increased out-of-phase polarization component (Fig. S6 in the supplementary material). Thus, an influence of the conductivity on the determined parameters (due to altered hysteresis shape) and/or the hardening mechanisms (due to charge migration) cannot be entirely excluded at the high-temperature end.

Although the $Eib$ and $Ec$ values at 20 °C are the highest for NBT–6BT–1Zn, they exhibit a more rapid decrease with temperature than those for NBT–6BT:10ZnO. This confirms the superior thermal stability of the composite hardening approach, which was already observed in the temperature dependence of $Q31R$ (Fig. 5). However, it must be pointed out that particularly in the case of composites, the relationship between $Eib/Ec$ and $Q31R$ is not trivial, in contrast to previous reports.^8,54 For example, while $Eib$ and $Ec$ of NBT–6BT:10ZnO feature severe thermally induced degradation, $Q31R$ exhibits pronounced thermal stability.

Harmonic analysis of the weak-field polarization response was carried out to examine the origin of the difference in temperature stability of the two hardening mechanisms. In particular, the amplitude of the third-harmonic $|P3|$⁠, in-phase $P′3$ and out-of-phase $P′′3$ components, and the phase angle $δ3=arctan(P′′3P′3)$ were determined as a function of electric field amplitude at different temperatures (Figs. 8 and S9 in the supplementary material). The higher harmonic amplitudes (⁠$|Pn|$ with n > 1) generally determine the nonlinearity of the polarization response, while specifically the third harmonics $P3′$ and $P3′′$ and their ratio $δ3$ are associated with in-phase (anhysteretic) and out-of-phase (hysteretic) nonlinear components, respectively, and $δ3$ determines their ratio.⁵⁵ Note that the remanent polarization of the poled samples remains mostly unchanged due to the subcoercive driving conditions $(E≤Ec)$⁠.⁵⁶ 

In ferroelectric and related materials, dielectric and electromechanical nonlinearities at the weak subcoercive electric field are mostly related to the non-lattice contributions. Depending on the material's nature, such contributions may have different origins,⁵⁷ including (i) domain wall motion, which is common to all ferroelectrics, (ii) interface boundary motion, particularly in MPB compositions,⁵⁸ (iii) PNR dynamics, typical in relaxor ferroelectrics,⁵⁹ or (iv) non-trivial interrelations of these. In NBT–6BT, where all these contributions may play a role, it is impossible to identify and deconvolute all contributions based on microscopic nonlinear harmonic analysis alone. Instead, we use an approach based on experimental data in hard and soft PZT combined with the implications of the Rayleigh model in the harmonic response.⁵⁶ In hard PZT, where aligned defect complexes strongly pin the domain walls and reduce their dynamics, the nonlinear response at the weak electric field is often characterized by $δ3=−180°$⁠, which is interpreted as an anhysteretic nonlinear response related to reversible movements of pinned domain walls. On the other hand, the Rayleigh model predicts that $δ3=−90°$⁠, which signifies that the nonlinear response is purely hysteretic and related to irreversible domain wall dynamics in a hypothetical energy potential with randomly distributed pinning centers. Thus, by combining experimental data and modeling, we assume here that the in-phase $P′3$ and out-of-phase $P′′3$ responses are related to reversible and irreversible nonlinear domain wall dynamics, respectively. The evolution of these third-harmonic parameters with electric field and temperature can be used as a fingerprint of the nonlinear dynamics of domain walls or similar interfaces in the material.⁵⁶ 

Qualitatively, all samples exhibit similar behavior of the third-harmonic parameters. Figures 8(a)–8(c) describe an electric field-dependent increase of the $|P3|$ amplitude (⁠$P3′$ and $P3′′$ amplitudes are given in Fig. S9 in the supplementary material), which increases with temperature and clearly reveals thermally activated nonlinear dynamic contributions to the polarization response. The third-harmonic phase angle $δ3$ evolves from around −180° at low electric fields toward −90° at large fields [Figs. 8(d)–8(f)]. Considering that all of the analyzed NBT–6BT compositions feature hardening characteristics (Fig. 7) and in analogy to the same behavior typically observed in hard PZT,⁵⁶ this evolution can be interpreted as a transition from an anhysteretic (reversible) response to a more hysteretic (irreversible) response with increasing electric field amplitude. Despite the same evolution in all compositions, however, the samples show distinct differences in field dependence. The nonlinear anhysteretic-to-hysteretic transition, determined from the $δ3(E)$ inflection point, is clearly lower in undoped NBT–6BT (0.2 kV/mm) than in NBT–6BT:10ZnO (0.49 kV/mm) and NBT–6BT–1Zn (0.73 kV/mm) and does not change with temperature. The lower electric field of the $δ3$ transition is consistent with the overall larger $|P3|$ and a larger increase in the first-harmonic permittivity, $ε1′$⁠, and loss angle, $tanδ1$⁠, with field (Fig. S8 in the supplementary material) in NBT–6BT as compared to the modified samples. The shift of the anhysteretic-to-hysteretic transition to a larger electric field in the modified compositions demonstrates an effective suppression of the hysteretic nonlinear response by Zn²⁺ doping and ZnO inclusions and correlates with the larger $Ec$⁠, $Eib$⁠, and $Q31R$ values, and the lower electromechanical coefficients (Figs. 5 and 7, Table I). However, the saturation values of $δ3$ are temperature-independent in NBT–6BT:10ZnO but strongly increase with temperature in NBT–6BT–1Zn. In other words, the ratio between the hysteretic and anhysteretic components of the nonlinear polarization response strongly increases with temperature only in NBT–6BT–1Zn. While more rigorous analysis should be performed, it is interesting to note the correlation between this temperature-dependent reversible-to-irreversible dynamic evolution with the pronounced decrease of $Q31R$ with temperature in NBT–6BT–1Zn [Fig. 5(a)].

Both Zn²⁺ doping and ceramic–ceramic composite formation with ZnO increase simultaneously the depolarization temperature $(Td)$ and the resonance quality factor $(QijR)$ of NBT–6BT. The depolarization process terminates directly in the transition of the ferroelectric phase into the ergodic relaxor phase without prior polarization detexturization; thus, a depolarization temperature increase to higher temperatures is realized by the shift of the ferroelectric-to-relaxor transition temperature $TF−R$⁠. However, the depolarization temperature $(Td)$ as well as the ferroelectric–relaxor transition temperature $(TF−R)$ are not simply shifted to a higher temperature on average but are spread out over a wide temperature regime. While the onset temperature for depolarization remains unaltered at about 86 °C as in undoped NBT–6BT, the final depolarization occurs between 130 and 133 °C and 117 and 120 °C for the composites and the doped materials, respectively. It is suggested that the broadening is associated with the induced increase of $TF−R$ of the tetragonal phase, while the $TF−R$ of the rhombohedral phase remains unaffected. The broadening reveals the decoupling of the piezoelectric $d33$ and $d31$ coefficients in the depolarization region, which was explained by a two-phase model, where the remanent textured ferroelectric phase is embedded in a growing ergodic relaxor phase.

The Zn²⁺-doped compositions displayed strong temperature dependence, similar to acceptor-doped PZT, indicating different contributions to $Q31R$ arising from acceptor doping. On the other hand, the NBT–6BT composite formation with ZnO inclusions leads to an overall smaller increase of the room temperature $Q31R$⁠, but the value is nearly temperature-invariant throughout the entire temperature range up to $Td$⁠. Polarization harmonic measurements have indicated a correlation between the thermally activated increase in the ratio of reversible-to-irreversible nonlinear dynamic contribution and the severe $Q31R$ degradation in Zn²⁺-doped NBT–6BT. The thermal stability of the NBT–6BT composites is consistent with the proposed hardening mechanism based on mechanical mismatch stresses, which are less effective in suppressing the irreversible nonlinear contributions but do not significantly alter the observed temperature range.

See the supplementary material for further details on the microstructural, dielectric, and piezoelectric properties of all compositions, the temperature-dependent ferroelectric loops, and the polarization harmonics analysis.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. 414073759 (KO 5100/3-1). J.K. additionally acknowledges the financial support from the Athene Young Investigator Program of the Technical University of Darmstadt. Part of the work was also supported by the DAAD through funds from the Bundesministerium für Bildung und Forschung (BMBF) under Grant No. 57402439 and the Slovenian Research Agency (SRA) through the bilateral project PR-08298 (Contract No. BI-DE/18-19-007). T.R. acknowledges core funding provided by SRA (No. P2-0105). L.K.V. acknowledges and thanks the Alexander von Humboldt Foundation for financial support.

The data that support the findings of this study are available within the article, its supplementary material, and from the corresponding author upon reasonable request.