Glassy Properties of the Lead-Free Isovalent Relaxor BaZr0.4Ti0.6O3

Glassy dielectric properties were investigated in lead-free BaZr0.4Ti0.6O3 (BZT40) ceramic samples using dielectric spectroscopy in the frequency range of 0.003 Hz–1 MHz and at temperatures of 10 K < T < 300 K. Measurements of the quasistatic dielectric polarization in bias electric fields up to ~28 kV/cm suggested that a ferroelectric state could not be induced, in contrast to the case of canonical relaxors such as PMN. The quasistatic dielectric and freezing dynamics results for the above field amplitudes showed that BZT40 effectively behaves as a dipolar glass. The relaxation spectrum was analyzed employing a frequency–temperature plot, which showed that the longest relaxation time obeyed the Vogel–Fulcher relation τ = τ0exp[E0/(T − T0)], with a freezing temperature of 76.7 K. The shortest relaxation time, in contrast, was characterized by a freezing temperature value close to 0 K, implying an Arrhenius-type behavior. The higher value of the polarization and the nonlinear third-order dielectric coefficient ε3 indicated a shift from a pseudospin glass behavior observed for BaZr0.5Ti0.5O3 (BZT50) toward a classical relaxor ferroelectric state.


Introduction
Since the 1960s, relaxor ferroelectrics have remained in the scientific focus because of their exceptional physical properties [1], such as high dielectric permittivity, large polarization and pyroelectricity, and significant electromechanical response [2][3][4][5][6][7]. Recently, it was shown that they also possess large caloric properties [8][9][10]. While the former properties benefit from the electric field-induced long-range ferroelectric order, an inherent property of relaxors in contrast to dipolar glasses, the electrocaloric response and wide temperature stability of these properties are enhanced by the disordered glassy nature of these materials [1][2][3][4]. Similarly to dipolar glasses at low electric fields, the glassy nature is reflected in the absence of a long-range ferroelectric order, i.e., spontaneous polarization and symmetry change at very low temperatures, formation of polar nanoregions (PNRs) appearing at relatively high temperatures, and broad dispersion of the complex dielectric permittivity. At the freezing temperature, the longest dipolar relaxation time diverges; hence, ergodicity breaking occurs [2,4]. Relaxors such as the disordered perovskite PbMg 1/3 Nb 2/3 O 3 (PMN) [3] thus exhibit dipolar glass properties below or ferroelectric properties above the critical field line E C (T) [2][3][4][5].
Theoretical first-principles studies and microscopic modeling [11,[24][25][26] of BZT with x = 0.50 indicated the formation of PNRs with a negligible contribution of random electric fields and strains and that BZT could be mapped to a soft pseudospin glass. The concentration x = 0.50 was experimentally investigated [15,27,28], raising arguments about whether BZT in the composition range of 0.25 < x < 0.75 exhibits properties of a lead-free isovalent relaxor or of a dipolar glass.
Our initial BZT50 dielectric spectroscopy study [27], through the Kutnjak or temperature -frequency plot, demonstrated the asymmetric stretching of the relaxation spectrum in which different portions of the spectrum followed the divergent Vogel-Fulcher relation; only the highest frequencies obeyed the Arrhenius equation describing the activated process. However, such behavior can be found for both relaxors and dipolar glasses. Several criteria can be found in the literature to discern between the relaxor and the dipolar glass states. (i) In dipolar glasses, the distribution of the relaxation times f (τ) is a simple function with a single peak as a consequence of the fact that PNRs in dipolar glasses will remain small, on the scale of a few unit cells. In contrast, PNRs in relaxors, due to their larger size and stronger coupling to the applied electric field, show breathing modes besides the flipping one. This could result in the second peak in f (τ). The shape of f (τ) can be extracted from dielectric dispersion data utilizing the so-called Tikhonov regularization method [29][30][31][32]. However, in many cases, both processes could be merged into a single peak distribution function, in which the lower frequency glassy tail diverges. At the same time, the high-frequency breathing part follows an activated process. (ii) Another criterion exploits the fact that in canonical relaxors like PMN, the electric field could induce a ferroelectric phase via a discontinuous transition [6,33], resulting in box-like hysteresis loops with sharp steps on rising slopes. In contrast, very narrow hysteresis loops can be found in dipolar glasses, and no critical field line E C (T) can be induced. (iii) The nonlinear third-order dielectric coefficient ε 3 is typically small in dipolar glasses propagating the linear dielectric response to higher fields. In contrast, classical relaxor systems exhibit a larger ε 3 , thus showing the nonlinear dielectric response already at modest electric fields.
To verify the impact of Zr reduction on the pseudospin glass state in BZT, the quasistatic and dynamic dielectric properties of BZT at x = 0.40 were experimentally determined. BZT40 possesses a lower concentration of Zr compared to the previously studied BZT50 ceramic and is thus closer to the lower limit of the relaxor regime approaching the ferroelectric one. The standard methods of monitoring the field-cooled quasistatic polarization on cooling, testing the nonlinearity of polarization loops as well as the possible existence of the critical field line E C (T), were employed to establish the nature of the ordering in BZT40 [34].

Materials and Methods
Powders of BZT with x = 0.40 were synthesized using the same route as that for the BZT50 samples, i.e., a solid-state reaction [35]. The precursors were calcinated in stoichiometric amounts at 1273 K for 4 h [35]. The powder was compacted at 1500 bar using cold isostatic pressing and sintered at 1773 K for 4 h [35]. The obtained ceramics with a relative density above 99% had a single-phase perovskite structure and a grain size of a few microns [18,27]. We cut 0.26 mm thick ceramics samples from pellets of 6 mm diameter, and gold electrodes of 5 mm diameter were sputtered on the pellet surfaces. The complex dielectric permittivity ε * ( f , T) = ε − iε was measured in the temperature interval from 10 K to 300 K and in the range of frequencies 0.003 Hz < f < 1 MHz by a Novocontrol Technologies Beta High Performance Impedance Analyzer capable to reliably measure the dielectric permittivity in the frequency range from 20 MHz to µHz. The peak-to-peak amplitude of the measuring AC electric field was 10 V/cm. An Oxford Instruments Limited continuous-flow cryostat operating between 1.5 K and 500 K and an ITC4 temperature controller were used to stabilize the temperature within ±0.1 K.
The quasistatic dielectric polarization P(T, E) was measured in field-cooled (FC) and zero-field-heated (ZFH) runs via the charge-collection technique [27,33]. The induced FC (or ZFH) polarization charge by various external dc bias fields (from 5 kV/cm to 28 kV/cm) was collected by a Keithley 6517B programmable Electrometer/High Resistance Meter [36]. The cooling (FC) or heating rate (ZFH) of 1.6 K/min was used. The FC polarization was measured on cooling in the external field down to 10 K, where the external field had been removed from the sample, and on heating, the ZFH quasistatic dielectric polarization was determined.
Virgin P(E) branches were determined by a homemade Sawyer-Tower bridge and a Keithley 6517B programmable electrometer. The period of the external AC electric field, changing from 0 kV/cm to 28 kV/cm and back, was approximately 500 s. The P(E) branches were measured at a specific temperature after annealing the sample at room temperature in the absence of the electric field to remove any history effects. Figures 1 and 2 show the ε and ε as functions of the temperature in the range of frequencies of 0.003 Hz-1 MHz. Such typical variation is found in dipolar glasses or relaxors with an increasing shift of the temperature position of the peak with decreasing frequency. Figure 3 shows the frequency f as a function of the temperature T at which ε ( f , T) achieved its maximum value. The frequency f was found to follow the Vogel-Fulcher (VF) relation

Results and Discussion
microns [18,27]. We cut 0.26 mm thick ceramics samples from pellets of 6 mm diameter, and gold electrodes of 5 mm diameter were sputtered on the pellet surfaces.
The complex dielectric permittivity  * ( , ) = ′ − ′′ was measured in the temperature interval from 10 K to 300 K and in the range of frequencies 0.003 Hz < f < 1 MHz by a Novocontrol Technologies Beta High Performance Impedance Analyzer capable to reliably measure the dielectric permittivity in the frequency range from 20 MHz to µHz. The peak-to-peak amplitude of the measuring AC electric field was 10 V/cm. An Oxford Instruments Limited continuous-flow cryostat operating between 1.5 K and 500 K and an ITC4 temperature controller were used to stabilize the temperature within ± 0.1 K. The quasistatic dielectric polarization ( , ) was measured in field-cooled (FC) and zero-field-heated (ZFH) runs via the charge-collection technique [27,33]. The induced FC (or ZFH) polarization charge by various external dc bias fields (from 5 kV/cm to 28 kV/cm) was collected by a Keithley 6517B programmable Electrometer/High Resistance Meter [36]. The cooling (FC) or heating rate (ZFH) of 1.6 K/min was used. The FC polarization was measured on cooling in the external field down to 10 K, where the external field had been removed from the sample, and on heating, the ZFH quasistatic dielectric polarization was determined.
Virgin ( ) branches were determined by a homemade Sawyer-Tower bridge and a Keithley 6517B programmable electrometer. The period of the external AC electric field, changing from 0 kV/cm to 28 kV/cm and back, was approximately 500 s. The ( ) branches were measured at a specific temperature after annealing the sample at room temperature in the absence of the electric field to remove any history effects.

Results and Discussion
Figures 1 and 2 show the ′ and  as functions of the temperature in the range of frequencies of 0.003 Hz-1 MHz. Such typical variation is found in dipolar glasses or relaxors with an increasing shift of the temperature position of the peak with decreasing frequency. Figure 3 shows the frequency as a function of the temperature at which  ( , ) achieved its maximum value. The frequency was found to follow the Vogel- Here, = 3.79 × 10 10 Hz is the attempt frequency, = 870 K represents the VF activation energy, and = 70.3 K ± 0.8 K is the VF freezing temperature [37]. The interpretation of as the freezing temperature in the static limit is not unique because ′ and  provided a slightly different value for , as the temperature positions of ( , ) and  ( , ) diffuse peaks did not match (see Figure 1).    A comparison of the dielectric data observed for BZT50 [27] and BZT40 showed that these two sets of data were qualitatively quite similar, with the dispersive temperature dependence of the real and imaginary part of dielectric permittivity whose peaks followed the Vogel-Fulcher relation. By lowering the Zr content from x = 0.5 to x = 0.4, the dielectric peaks were shifted to higher temperatures for about 60 K, increasing the dielectric values by about 50%. Consequently, the increased from 9.5 K to 70.3 K, the attempt frequency decreased of about three orders of magnitude (4.08 × 10 13 Hz to 3.79 × 10 10 Hz), while the VF activation energy decreased from 2255 K to 870 K. These observations indicated the approach of the ferroelectric regime as the Zr content decreased.
The so-called Cole-Cole diagram ( Figure 4) in which  is parametrically plotted as a function of  could reveal the static and high-frequency limits of the dielectric relaxation as well as the asymmetry of the relaxation spectrum. The increasing asymmetric shape and strong suppression of the dispersion with the decreasing temperature indicated a strong asymmetric broadening of the relaxation spectrum. The Havriliak-Negami (HN) function was employed to extract the static dielectric permittivity,  , and the high-frequency limit, ,   A comparison of the dielectric data observed for BZT50 [27] and BZT40 showed that these two sets of data were qualitatively quite similar, with the dispersive temperature dependence of the real and imaginary part of dielectric permittivity whose peaks followed the Vogel-Fulcher relation. By lowering the Zr content from x = 0.5 to x = 0.4, the dielectric peaks were shifted to higher temperatures for about 60 K, increasing the dielectric values by about 50%. Consequently, the increased from 9.5 K to 70.3 K, the attempt frequency decreased of about three orders of magnitude (4.08 × 10 13 Hz to 3.79 × 10 10 Hz), while the VF activation energy decreased from 2255 K to 870 K. These observations indicated the approach of the ferroelectric regime as the Zr content decreased.
The so-called Cole-Cole diagram ( Figure 4) in which  is parametrically plotted as a function of  could reveal the static and high-frequency limits of the dielectric relaxation as well as the asymmetry of the relaxation spectrum. The increasing asymmetric shape and strong suppression of the dispersion with the decreasing temperature indicated a strong asymmetric broadening of the relaxation spectrum. The Havriliak-Negami (HN) function was employed to extract the static dielectric permittivity,  , and the high-frequency limit, , Figure 3. The frequency f as a function of the temperature T at which ε ( f , T) exhibits its maximum value (solid circles). The solid line is a fitting curve using Equation (1).
Here, f 0 =3.79 × 10 10 Hz is the attempt frequency, E 0 =870 K represents the VF activation energy, and T 0 = 70.3 K ± 0.8 K is the VF freezing temperature [37]. The interpretation of T 0 as the freezing temperature T F in the static limit is not unique because ε and ε provided a slightly different value for T 0 , as the temperature positions of ε ( f , T) and ε ( f , T) diffuse peaks did not match (see Figure 1).
A comparison of the dielectric data observed for BZT50 [27] and BZT40 showed that these two sets of data were qualitatively quite similar, with the dispersive temperature dependence of the real and imaginary part of dielectric permittivity whose peaks followed the Vogel-Fulcher relation. By lowering the Zr content from x = 0.5 to x = 0.4, the dielectric peaks were shifted to higher temperatures for about 60 K, increasing the dielectric values by about 50%. Consequently, the T 0 increased from 9.5 K to 70.3 K, the attempt frequency f 0 decreased of about three orders of magnitude (4.08 × 10 13 Hz to 3.79 × 10 10 Hz), while the VF activation energy E 0 decreased from 2255 K to 870 K. These observations indicated the approach of the ferroelectric regime as the Zr content decreased.
The so-called Cole-Cole diagram ( Figure 4) in which ε is parametrically plotted as a function of ε could reveal the static and high-frequency limits of the dielectric relaxation as well as the asymmetry of the relaxation spectrum. The increasing asymmetric shape and strong suppression of the dispersion with the decreasing temperature indicated a strong asymmetric broadening of the relaxation spectrum. The Havriliak-Negami (HN) function was employed to extract the static dielectric permittivity, ε S , and the high-frequency limit, ε ∞ , where ω = 2π f , and τ is the characteristic relaxation time. Examples of the fits are shown in the inset of Figure 4, with the fitting parameters presented in Table 1.
where  = 2 , and  is the characteristic relaxation time. Examples of the fits are shown in the inset of Figure 4, with the fitting parameters presented in Table 1.  Table 1.  Figure 5 shows the static  and high-frequency limit fitting parameters as functions of the temperature. As the dielectric dispersion could not be covered with experimentally available frequencies, the fitting procedure allowed the estimation of the temperature dependence of , , and  in a relatively narrow range of temperatures. Fortunately, it was shown before that the static dielectric permittivity  coincides with the quasistatic dielectric value  , determined in the FC run at a low electric field [27,37,38].   Table 1. Table 1. The fitting parameters for the fitting curves shown in the inset of Figure 4.  Figure 5 shows the static ε S and high-frequency limit ε ∞ fitting parameters as functions of the temperature. As the dielectric dispersion could not be covered with experimentally available frequencies, the fitting procedure allowed the estimation of the temperature dependence of ε s , ε ∞ , and τ in a relatively narrow range of temperatures. Fortunately, it was shown before that the static dielectric permittivity ε S coincides with the quasistatic dielectric value ε FC , determined in the FC run at a low electric field [27,37,38].
where  = 2 , and  is the characteristic relaxation time. Examples of the fits are shown in the inset of Figure 4, with the fitting parameters presented in Table 1.  Table 1.  Figure 5 shows the static  and high-frequency limit fitting parameters as functions of the temperature. As the dielectric dispersion could not be covered with experimentally available frequencies, the fitting procedure allowed the estimation of the temperature dependence of , , and  in a relatively narrow range of temperatures. Fortunately, it was shown before that the static dielectric permittivity  coincides with the quasistatic dielectric value  , determined in the FC run at a low electric field [27,37,38].   The temperature-frequency plot introduced in Ref. [37] allows the observation of how various segments of the relaxation times distribution function vary with the temperature. The data analysis procedure via the temperature-frequency plot is described in detail in Refs. [37][38][39]. The temperature-frequency plot exploits a special presentation of the real part of the complex dielectric permittivity that provides direct information on the temperature evolution of the relaxation spectrum. In this method, it is presumed that the relaxation spectrum is broad but limited, i.e., constrained between the upper and the lower relaxation time cut-offs. The special presentation of the real part of the complex dielectric permittivity is introduced through the parameter δ = δ(ω, T) or the reduced dielectric permittivity, defined as where z = log(ω a τ), ω a is an arbitrary unit frequency, and z 1 and z 2 represent the limits of the distribution of the relaxation times [38,39]. The right part of Equation (3) can be considered the filter 1/1 + (ω/ω a ) 2 exp(2z) acting upon the distribution of the relaxation times g(z) that opens at z ≈ ln(ω a /ω). In the case of a broad distribution of the relaxation times, the value of the integral in Equation (3) is determined for low values of ε (ω, T) or δ ≈ 0 by contributions stemming from the short τ (high-frequency) part of g(z). In contrast, for values of ε close to the static dielectric permittivity ε s or δ ≈ 1, the whole distribution of the relaxation times g(z) contributes to the integral in Equation (3); thus, the frequency filter probes the shape of g(z) near its long relaxation time (low frequency) cut-off z 2 . Therefore, by scanning δ from 0 to 1, the filter in the second part of Equation (3) probes the distribution of the relaxation times g(z) by shifting its position in τ space. The position of the frequency filter can be used as a probe to discern the changes in the shape or polydispersity of the relaxation spectrum. Consequently, δ, which can be estimated directly from the dielectric permittivity data, actually represents contributions of g(z) for relaxation times below 1/ω. In the analysis, δ is taken as an independent parameter whose value is set to some fixed values between 0 and 1. Technically, for the chosen set of fixed values of δ at a given temperature, the corresponding set of frequency values f = ω/2π was determined by a suitable interpolation technique between the discrete experimental points at which a given ε was realized. It should be stressed again that scanning ε between ε S and ε ∞ changed uniquely δ from 1 to 0. The set of frequencies can also be determined from Figure 1 as cross sections between the experimental data ε (ω, T) curves and the ε = const ( δ = const) lines. Figure 6 presents ln( f ) vs. 1/T for δ values in the range of 0.05 < δ < 0.98.
Crystals 2023, 13, x FOR PEER REVIEW 6 of 12 The temperature-frequency plot introduced in Ref. [37] allows the observation of how various segments of the relaxation times distribution function vary with the temperature. The data analysis procedure via the temperature-frequency plot is described in detail in Refs. [37][38][39]. The temperature-frequency plot exploits a special presentation of the real part of the complex dielectric permittivity that provides direct information on the temperature evolution of the relaxation spectrum. In this method, it is presumed that the relaxation spectrum is broad but limited, i.e., constrained between the upper and the lower relaxation time cut-offs. The special presentation of the real part of the complex dielectric permittivity is introduced through the parameter  = (, ) or the reduced dielectric permittivity, defined as where = log( ),  is an arbitrary unit frequency, and and represent the limits of the distribution of the relaxation times [38,39]. The right part of Equation (3) can be considered the filter 1 1 + ( / ) exp (2 ) ⁄ acting upon the distribution of the relaxation times ( ) that opens at ≈ ln( / ). In the case of a broad distribution of the relaxation times, the value of the integral in Equation (3) is determined for low values of ´(, ) or  ≈ 0 by contributions stemming from the short  (high-frequency) part of ( ). In contrast, for values of ´ close to the static dielectric permittivity or  ≈ 1, the whole distribution of the relaxation times ( ) contributes to the integral in Equation (3); thus, the frequency filter probes the shape of ( ) near its long relaxation time (low frequency) cut-off . Therefore, by scanning  from 0 to 1, the filter in the second part of Equation (3) probes the distribution of the relaxation times ( ) by shifting its position in  space.
The position of the frequency filter can be used as a probe to discern the changes in the shape or polydispersity of the relaxation spectrum. Consequently, , which can be estimated directly from the dielectric permittivity data, actually represents contributions of ( ) for relaxation times below 1/. In the analysis,  is taken as an independent parameter whose value is set to some fixed values between 0 and 1. Technically, for the chosen set of fixed values of δ at a given temperature, the corresponding set of frequency values = /2 was determined by a suitable interpolation technique between the discrete experimental points at which a given  was realized. It should be stressed again that scanning  between  and changed uniquely  from 1 to 0. The set of frequencies can also be determined from Figure    The temperature variation of different parts of the relaxation spectrum related to various δ could be extracted by fitting each curve with a VF ansatz where δ runs over values representing each curve in Figure 6 [37]. Thus, the obtained freezing temperatures for different parts of the relaxation spectrum, i.e., the different values of δ are depicted in Figure 7. The longest 1/ f 2 and the shortest relaxation time 1/ f 1 correspond to the limiting cases δ → 1 and δ → 0, respectively. Figure 7 also shows T 0 vs. δ in BZT50 [27]. The fitting parameters for the relaxation spectrum cut-off frequencies f 1 and f 2 are listed in Table 2 for both BZT40 and BZT50 ceramics. It should be noted that, according to the experimental time scale, the value T 0 = 70.3 K obtained from the dispersion of ε ( f , T) peaks (see Figure 3) roughly matched the freezing temperature denoted by δ ≈ 0.95. It should be noted that the above analysis was possible only in the temperature range in which ε S and ε ∞ were well established.
cies and are listed in Table 2 for both BZT40 and BZT50 ceramics. It should be noted that, according to the experimental time scale, the value = 70.3 K obtained from the dispersion of ( , ) peaks (see Figure 3) roughly matched the freezing temperature denoted by  ≈ 0.95. It should be noted that the above analysis was possible only in the temperature range in which  and were well established. The impact of freezing dynamics is best manifested in the stark difference between FC and ZFH polarization [2,37]. The splitting of the quasistatic FC and ZFH (after fieldcooling) polarization is shown in Figure 8, measured at the dc electric field of 28 kV/cm. The difference between the FC and the ZFH curves at the lowest temperatures is due to the fast high-frequency relaxation of the polarization, i.e., to fast processes contributing to . From = ( − )/ , the dielectric dispersion high-frequency limit = 90 could be estimated in agreement with the HN function analysis value (see Figure 5). or the remanent polarization was detected after removing the external electric field at 10 K, after the FC experiment.   Freezing temperature T 0 vs. δ for BZT40 (red circles). The solid line is an extrapolation to δ → 1, corresponding to the limit of the longest relaxation times. The blue circles represent T 0 (δ) in BZT50 [27]. The impact of freezing dynamics is best manifested in the stark difference between FC and ZFH polarization [2,37]. The splitting of the quasistatic FC and ZFH (after fieldcooling) polarization is shown in Figure 8, measured at the dc electric field of 28 kV/cm. The difference between the FC and the ZFH curves at the lowest temperatures is due to the fast high-frequency relaxation of the polarization, i.e., to fast processes contributing to ε ∞ . From ε ∞ = (P FC − P ZFH )/ε ∞ E, the dielectric dispersion high-frequency limit ε ∞ = 90 could be estimated in agreement with the HN function analysis value (see Figure 5). P ZFH Crystals 2023, 13, 1303 8 of 12 or the remanent polarization P r was detected after removing the external electric field at 10 K, after the FC experiment. tion spectrum shifted back into the experimental time window. relaxed fast toward the longest relaxation time's freezing temperature = 74.4 K, at which the steepest slope in ( ) variation changed, followed by a slowly decreasing shoulder, which vanished at ≈225 K (see Figure 8). In contrast to , the freezing temperature ≈ 50 K depends on the experimental time scale and cannot be considered a true static quantity. The shift of to higher temperatures and the long polarization tail persisting to high temperatures ( Figure 8) indicated the shift of the BZT40 state from a pseudospin-glass state [9] toward the classical relaxor-type behavior. A similar shift was found recently in another isovalent relaxor system, BaTi1-xCexO3, where the composition x = 0.30 showed dipolar glass properties in contrast to the composition x = 0.20 with relaxor ferroelectric properties [40,41]. To investigate the nonlinear properties of the induced polarization ( ), the quasistatic electric external field was cycled from 0 kV/cm up to 28 kV/cm at a low frequency of ≈0.002 Hz. In a typical relaxor, like PMN, "slim" nonlinear polarization loops with negligibly small remanent polarization are usually observed in a portion of the electric fieldtemperature ( − ) phase diagram near or for temperatures above the freezing line (dashed line in Figure 9) [33].  The quasistatic FC (blue curve) and ZFH (or P r measured after a field-cooled run) (red curve) polarization as a function of temperature. The FC curve was obtained in the 28 kV/cm electric field. The temperature T 02 = 74.4 K corresponds to the temperature of the knee in the P r (T) slope variation. Figure 8 shows that P ZFH or P r were almost unchanged to the temperature T F ≈ 50 K, after which the P r rapidly started to decrease because, above 50 K, most of the relaxation spectrum shifted back into the experimental time window. P r relaxed fast toward the longest relaxation time's freezing temperature T 02 = 74.4 K, at which the steepest slope in P r (T) variation changed, followed by a slowly decreasing shoulder, which vanished at ≈225 K (see Figure 8). In contrast to T 02 , the freezing temperature T F ≈ 50 K depends on the experimental time scale and cannot be considered a true static quantity. The shift of T 02 to higher temperatures and the long polarization tail persisting to high temperatures (Figure 8) indicated the shift of the BZT40 state from a pseudospin-glass state [9] toward the classical relaxor-type behavior. A similar shift was found recently in another isovalent relaxor system, BaTi 1-x Ce x O 3 , where the composition x = 0.30 showed dipolar glass properties in contrast to the composition x = 0.20 with relaxor ferroelectric properties [40,41].
To investigate the nonlinear properties of the induced polarization P(E), the quasistatic electric external field was cycled from 0 kV/cm up to 28 kV/cm at a low frequency of ≈0.002 Hz. In a typical relaxor, like PMN, "slim" nonlinear polarization loops with negligibly small remanent polarization are usually observed in a portion of the electric field-temperature (E − T) phase diagram near or for temperatures above the freezing line (dashed line in Figure 9) [33].
Crystals 2023, 13, x FOR PEER REVIEW 8 of 12 Figure 8 shows that or were almost unchanged to the temperature ≈ 50 K, after which the rapidly started to decrease because, above 50 K, most of the relaxation spectrum shifted back into the experimental time window.
relaxed fast toward the longest relaxation time's freezing temperature = 74.4 K, at which the steepest slope in ( ) variation changed, followed by a slowly decreasing shoulder, which vanished at ≈225 K (see Figure 8). In contrast to , the freezing temperature ≈ 50 K depends on the experimental time scale and cannot be considered a true static quantity. The shift of to higher temperatures and the long polarization tail persisting to high temperatures ( Figure 8) indicated the shift of the BZT40 state from a pseudospin-glass state [9] toward the classical relaxor-type behavior. A similar shift was found recently in another isovalent relaxor system, BaTi1-xCexO3, where the composition x = 0.30 showed dipolar glass properties in contrast to the composition x = 0.20 with relaxor ferroelectric properties [40,41]. To investigate the nonlinear properties of the induced polarization ( ), the quasistatic electric external field was cycled from 0 kV/cm up to 28 kV/cm at a low frequency of ≈0.002 Hz. In a typical relaxor, like PMN, "slim" nonlinear polarization loops with negligibly small remanent polarization are usually observed in a portion of the electric fieldtemperature ( − ) phase diagram near or for temperatures above the freezing line (dashed line in Figure 9) [33].  At temperatures below the freezing line, "square" loops were observed at fields above the critical threshold to induce the ferroelectric phase [33]. In the region above the freezing temperature line T F (E) (dashed line in Figure 9), polar nanoclusters appeared as fluctuating entities, and P r remained nearly zero; however, P(E) remained strongly nonlinear [33,42]. By contrast, the field could not induce large domains in dipolar glasses, and single dipoles or very small dipolar clusters remained the only relaxing entities. Such a state resembled a paraelectric state characterized by linear properties, with a small third-order nonlinear dielectric coefficient, ε 3 . The remanent polarization resembled frozen polar clusters oriented in the external electric field, which, below the freezing temperature, remained ordered even after the field was turned off. Figure 10 shows the induced polarization P(E) as a function of the applied field at different temperatures above the freezing temperature of 74.4 K. The polarization was measured by increasing and subsequently decreasing the field from 0 to 28 kV/cm within the 500 s experimental time scale. The application of a negative electric field reproduced symmetric negative polarization branches. The experimental time scale could be considered quasistatic in the temperature range above 80 K. Here, quasistatic "loops" did not show any hysteresis, as shown in Figure 10, where both up and down P(E) data lie on the same curve, i.e., the remanent polarization is zero. Due to the increasing longest relaxation times at lower temperatures, the data obtained when decreasing the field could not relax within the experimental time scale back to the initial values, resulting in artificial dynamic hysteresis. It was verified that this dynamic hysteresis disappeared if the time scale of the experiment increased. Verification was carried out up to the 2000 sec time scale. The observed maximum polarization of 10 µC/cm 2 at 28 kV/cm exceeded almost by a factor of 2 the polarization observed in the same field of 28 kV/cm in BZT50 (5.5 µC/cm 2 ). The nonlinearity of the P(E) = ε 0 ε 1 E + ε 3 E 3 + ε 5 E 5 was strong, with the absolute value of the otherwise negative third-order nonlinear dielectric coefficient −ε 3 = 2×10 −21 AsV −3 m at 225 K increasing toward −ε 3 = 4×10 −21 AsV −3 m at 80 K, i.e., with the temperature approaching the freezing temperature. The absolute value of the third-order nonlinear dielectric coefficient ε 3 for BZT40 was about one order of magnitude larger than that for BZT50 [27]. The nonlinearity increased with the decreasing temperature, indicating the approaching onset of the long-range ferroelectric order. The experimentally limited maximum field of 28 kV/cm was insufficient to induce a ferroelectric phase, and the question remains whether a long-range ferroelectric order could be induced in BZT40. At temperatures below the freezing line, "square" loops were observed at fields above the critical threshold to induce the ferroelectric phase [33]. In the region above the freezing temperature line ( ) (dashed line in Figure 9), polar nanoclusters appeared as fluctuating entities, and remained nearly zero; however, ( ) remained strongly nonlinear [33,42]. By contrast, the field could not induce large domains in dipolar glasses, and single dipoles or very small dipolar clusters remained the only relaxing entities. Such a state resembled a paraelectric state characterized by linear properties, with a small thirdorder nonlinear dielectric coefficient, . The remanent polarization resembled frozen polar clusters oriented in the external electric field, which, below the freezing temperature, remained ordered even after the field was turned off. Figure 10 shows the induced polarization ( ) as a function of the applied field at different temperatures above the freezing temperature of 74.4 K. The polarization was measured by increasing and subsequently decreasing the field from 0 to 28 kV/cm within the 500 s experimental time scale. The application of a negative electric field reproduced symmetric negative polarization branches. The experimental time scale could be considered quasistatic in the temperature range above 80 K. Here, quasistatic "loops" did not show any hysteresis, as shown in Figure 10, where both up and down ( ) data lie on the same curve, i.e., the remanent polarization is zero. Due to the increasing longest relaxation times at lower temperatures, the data obtained when decreasing the field could not relax within the experimental time scale back to the initial values, resulting in artificial dynamic hysteresis. It was verified that this dynamic hysteresis disappeared if the time scale of the experiment increased. Verification was carried out up to the 2000 sec time scale. The observed maximum polarization of 10 µC/cm 2 at 28 kV/cm exceeded almost by a factor of 2 the polarization observed in the same field of 28 kV/cm in BZT50 (5.5 µC/cm 2 ). The nonlinearity of the ( ) = + + was strong, with the absolute value of the otherwise negative third-order nonlinear dielectric coefficient − = 2 × 10 −21 AsV −3 m at 225 K increasing toward − = 4 × 10 −21 AsV −3 m at 80 K, i.e., with the temperature approaching the freezing temperature. The absolute value of the third-order nonlinear dielectric coefficient for BZT40 was about one order of magnitude larger than that for BZT50 [27]. The nonlinearity increased with the decreasing temperature, indicating the approaching onset of the long-range ferroelectric order. The experimentally limited maximum field of 28 kV/cm was insufficient to induce a ferroelectric phase, and the question remains whether a long-range ferroelectric order could be induced in BZT40.    Figure 11 shows the temperature dependence of the linear dielectric permittivity ε ( f , T) determined in the range of frequencies from 0.1 Hz to 1 MHz in a low electric field of 10 V/cm and ε FC , obtained in an FC run with a dc field of 5 kV/cm, below which P(E) is still linear (see Figure 10), thus allowing the calculation of the linear dielectric permittivity ε FC = P FC /ε 0 E from the measured P FC . The static dielectric permittivity obtained from HN function fits (red circles) agreed well with the quasistatic ε FC obtained from the FC experiments, thus allowing the determination of the static dielectric permittivity at temperatures below the freezing temperature, i.e., in the temperature range experimentally inaccessible by established dielectric techniques.   Figure 5). The red curve represents ZFH or measured after a field-cooled run.

Conclusions
This dielectric spectroscopy study was performed to verify the dynamic and quasistatic dielectric properties of isovalent relaxor BZT40 ceramics. Dielectric spectroscopy in the frequency range 0.003 Hz-1 MHz and at temperatures of 10 K < T < 300 K revealed, via temperature-frequency plot analysis, that different parts of the relaxation spectrum followed the Vogel-Fulcher law. Only the very high-frequency part behaved according to the Arrhenius-activated process. The maximum relaxation time was found to diverge at the freezing temperature of 74.4 K. Such features were observed in both pseudospin glass and relaxor ferroelectric systems. The quasistatic dielectric polarization P(E) measured in an FC experiment was about two times larger than that of BZT50, measured in the same field of 28 kV/cm. Consequently, the quasistatic dielectric permittivity  = 5300, measured in the linear regime, was also twice larger than in BZT50. In addition, the polarization field dependence P(E) was more strongly nonlinear than in BZT50, with the absolute value of the third-order nonlinear dielectric coefficient = −4 × 10 −21 AsV −3 m being ten times larger than that of BZT50. Up to 28 kV/cm, however, no transition to a ferroelectric-ordered state could be induced. The higher dielectric permittivity P(E) and the third-order dielectric nonlinearity suggest that BZT40 ceramics exhibit a crossover from the incipient relaxor state, i.e., the state closer to the pseudospin glass found in BZT50 ceramics, towards the diffuse ferroelectric (relaxor) state observed in BZT compositions with 0 < x ≤ 0.2. The high values of the above physical quantities observed in BZT40 are typical properties of classical relaxors like PMN. In contrast to PMN and similar to BZT50, the electric field could not induce the ferroelectric state and had no substantial influence on dipolar dynamics in BZT40 ceramics.

Conclusions
This dielectric spectroscopy study was performed to verify the dynamic and quasistatic dielectric properties of isovalent relaxor BZT40 ceramics. Dielectric spectroscopy in the frequency range 0.003 Hz-1 MHz and at temperatures of 10 K < T < 300 K revealed, via temperature-frequency plot analysis, that different parts of the relaxation spectrum followed the Vogel-Fulcher law. Only the very high-frequency part behaved according to the Arrhenius-activated process. The maximum relaxation time was found to diverge at the freezing temperature of 74.4 K. Such features were observed in both pseudospin glass and relaxor ferroelectric systems. The quasistatic dielectric polarization P(E) measured in an FC experiment was about two times larger than that of BZT50, measured in the same field of 28 kV/cm. Consequently, the quasistatic dielectric permittivity ε FC = 5300, measured in the linear regime, was also twice larger than in BZT50. In addition, the polarization field dependence P(E) was more strongly nonlinear than in BZT50, with the absolute value of the third-order nonlinear dielectric coefficient ε 3 = −4 × 10 −21 AsV −3 m being ten times larger than that of BZT50. Up to 28 kV/cm, however, no transition to a ferroelectric-ordered state could be induced.
The higher dielectric permittivity P(E) and the third-order dielectric nonlinearity ε 3 suggest that BZT40 ceramics exhibit a crossover from the incipient relaxor state, i.e., the state closer to the pseudospin glass found in BZT50 ceramics, towards the diffuse ferroelectric (relaxor) state observed in BZT compositions with 0 < x ≤ 0.2. The high values of the above physical quantities observed in BZT40 are typical properties of classical relaxors like PMN. In contrast to PMN and similar to BZT50, the electric field could not induce the ferroelectric state and had no substantial influence on dipolar dynamics in BZT40 ceramics.
Funding: This work was supported by the Slovenian Research and Innovation Agency under the programs P1-0125 and P1-0044.

Data Availability Statement:
The data that support the findings of the study are available within this article.