Electrical Studies of Barkhausen Switching Noise in Ferroelectric lead zirconate titanate (PZT) and BaTiO3: Critical Exponents and Temperature-dependence

TL;DR

This study investigates Barkhausen noise in ferroelectric materials PZT and BaTiO3, focusing on critical exponents and temperature effects, revealing new temperature-dependent behaviors and critical exponents in these materials.

Contribution

It provides the first measurement of the Omori exponent in PZT and extends temperature-dependent studies of Barkhausen noise in BaTiO3.

Findings

Omori exponent in PZT: -0.95±0.03

Temperature dependence of Barkhausen noise in BaTiO3

Agreement with avalanche models in energy spectrum

Abstract

Previous studies of Barkhausen noise in PZT have been limited to the energy spectrum (slew rate response voltages versus time), showing agreement with avalanche models; in barium titanate other exponents have been measured acoustically, but only at ambient temperatures. In the present study we report the Omori exponent (-0.95$\pm$0.03) for aftershocks in PZT and extend the barium titanate studies to a wider range of temperature.

Tables

Table 1. Table 1: Previous studies of Barkhausen pulses in ferroelectrics and ferroelastics 2 2 2 See also: D. Lupascu, Fatigue in Ferroelectric Ceramics and Related Issues (Springer, Berlin 2013)

Material	References
LiTaO3, LiNbO3, Gd2(MoO4)3	a) b)
PZT	b) c)
PbTiO3	d) e) f)
Ferroelastic Pb2(PO4)3	g)
BaTiO3	h) k)
Pb5Ge3O11	i)
PbScTaO3- and Bi0.5Na0.5TiO3- mixed alloys	j)
Pb(Mg1/3Ta2/3)O3 , PbFe1/2Nb1/2O3	k)
BST (BaxSr1-xTiO3)	l)

Table 2. Table 2: Summary of linear fitting to the three exponents seen in the PLH for PZT at room temperature with 100 measurements (Figure 4 )

Exponent	Value
${ϵ}_1$	$1.47\pm 0.04$
${ϵ}_2$	$1.79\pm 0.08$
${ϵ}_3$	$2.04\pm 0.19$

Table 3. Table 3: Summary of energy exponents obtained from power-law histograms at various temperatures.

Temperature, K	${ϵ}_1$	${ϵ}_2$	${ϵ}_3$
160	1.29 $\pm$ 0.10	2.10 $\pm$ 0.21
227	1.26 $\pm$ 0.21	1.68 $\pm$0.18
259	1.30 $\pm$ 0.06	1.80 $\pm$ 0.12
323	1.27 $\pm$ 0.17	2.19 $\pm$0.17	2.23 $\pm$ 0.61
333	1.67 $\pm$ 0.13	1.92 $\pm$ 0.19
353	1.54 $\pm$ 0.11	1.92 $\pm$ 0.35
373	0.99 $\pm$ 0.36	2.01 $\pm$ 0.54	2.32 $\pm$ 0.43
393	1.24 $\pm$ 0.08	2.64 $\pm$ 0.25
413	1.20 $\pm$ 0.10	2.33 $\pm$ 0.33
423	1.48 $\pm$ 0.14	2.06 $\pm$ 0.64	2.52 $\pm$ 0.21
433	1.03 $\pm$ 0.18	2.09 $\pm$ 0.21	2.30 $\pm$ 0.24
453	1.31 $\pm$ 0.25	1.81 $\pm$ 0.09
473	1.48 $\pm$ 0.14	1.9 $\pm$ 0.13

Equations

$$P[X]=\frac{\alpha-1}{X_{min}}\Big{(}\frac{X_{i}}{X_{min}}\Big{)}^{-\alpha}$$

$$n(t)=\frac{k}{c+t}$$

$$n(t)=\frac{k}{(c+t{)}^p}$$

$$P(J)\propto J^{-ϵ}$$

$$J\sim \int v(t{)}^{2}dt$$

$$P[X_i]=\frac{\tilde{P}[X_i]}{\exp (X_i)-\exp (X_{i+1})}$$

Full text

I Introduction

Recent studies of crepitation (crackling noise) of ferroelectric lead zirconate titanate (PZT) and barium titanate (BTO) samples during ferroelectric switching was limited to a single exponent (energy versus time) for PZT and a single temperature (ambient) for BTO. In the present study, using slew-rate data on voltage pulse output, we extend the PZT work to a wide temperature range above and below room temperature, and measure additional exponents, including both amplitude and aftershock (1894 Omori earthquake frequency) parameters. The data are compatible with avalanche models.

Ferroelectric and piezoelectric materials are used extensively in our day-to-day lives: in smart phones alone there are piezoelectric microphones and speakers, not to mention the touch screen itself; and in MRI medical imaging equipment PZT is a usual detector element, limited by electrical noise. There are also ferroelectric memory systems (FRAM) present in smart cards as well as capacitors in every component of the electronics. The understanding and application of these materials are both at an advanced stage, with in-depth knowledge of its drawbacks being vital in selecting the right material for the given problem. There has been extensive research carried out into tackling some of the more obvious drawbacks such as fatigue, retention and electrical breakdown Scott and De Araujo (1989); Damjanovic et al. (1998); Dawber et al. (2005). However, the noise and discontinuous nature of the ferroelectric response is still poorly understood. If the research were to continue on the same path, the small noise effects would become the limiting factors in the future, hence the importance of this investigation.

Additionally, some properties of noise in ferroelectrics have been shown to obey a power-law and fit into a universality class Tan et al. (2019); Dahmen and Ben-Zion (2009); Salje et al. (2017, 2019). This is particularly interesting when applied to the study of earthquakes. Earthquakes are not reproducible, but it has been found that the compression of porous materials have the same universality class Baró et al. (2013), so it is possible to map the physics of earthquakes onto this experiment in a way that is both easily reproducible and safe. Therefore, it is important to strengthen the link to earthquake statistics. The mechanism for polarisation in perovskite materials is well understood Scott and De Araujo (1989); Auciello et al. (1998); Rabe et al. (2007); Ahn et al. (2004); Schmidt (1967); Lines and Glass (2001); Setter et al. (2006). There is a qualitative parameter referred to as the hardness of the ferroelectric/ferromagnetic response Auciello et al. (1998); Setter et al. (2006); Altpeter et al. (2016); Chilibon and Marat-Mendes (2012); Guyonnet (2014). PZT shows a typical hard ferroelectric response (small coercive field; narrow hysteresis loop), and BTO shows a typical soft response.

There are several particular questions to address: For example, in the simplest kind of avalanche (sand-piles) the particles are spherical and neither their shape nor mobility changes with temperature. They typically exhibit single exponents over a wide range of measurements. However, in a ferroelectric, the domains are more often planar; they greatly decrease in size and increase in mobility as phase transition temperatures are approached; and their anisotropy is more like rice grains than sand grains. Does this affect their characteristic exponents? Second, the Omori (1984) model of earthquakes gives an exponent for the time dependence of aftershocks, but not for their intensities. Are these related? The magnitude of shocks in our Barkhausen experiments is ca. a few attoJoules, roughly 30 orders of magnitude lower than in some earthquakes (Richter scale 5.0 = 1012 J). It is rare in physics to find phenomena that scale over 30 orders of magnitude.

The maximum likelihood statistics used in this report can differentiate in a robust way between one exponent and a superposition of two. In most cases we find two exponents describe the power spectra (based upon slew rates). We tentatively interpret those as depinning from (1) point defects and (2) extended defects (such as threading dislocations, known to be in high concentration in PZT).

Barkhausen pulses have been studied previously in PZT and in other ferroelectrics and ferroelastics, particularly by three groups: Shur in Ekaterinburgh; Dul’kin and Roth in Israel; and Lupascu in Duisburg-Essen. Table I lists some of the earlier studies, which generally did not emphasise power-law exponents.

1. a)

V. Ya. Shur, E. L. Rumyantsev, D. V. Pelegov, V. L. Kozhevnikov, E. V. Nikolaeva, E. L. Shishkin, A. P. Chernykh & R. K. Ivanov, Ferroelectrics 267, 347 (2010); this references also cites 9 related papers by Shur et al.; see especially I. S. Baturin, M. V. Konev, A. R. Akhmatkhanov, A. I. Lobov, & V. Ya. Shur, Ferroelectrics 374, 136 (2008). 2. b)

Doru C. Lupascu , Jürgen Nuffer & Jürgen Rödel, Ferroelectrics 290, 203 (2011); D. Lupascu, T. Utschig, V. Ya. Shur & A. G. Shur, Ferroelectrics 290, 207 (2011). 3. c)

Jürgen Nuffer , Doru C. Lupascu & Jürgen Rödel , Ferroelectrics 240, 1293 (2000). 4. d)

Hideo Iwasaki & Mamoru Izumi, Ferroelectrics 37, 563 (2011). 5. e)

E. Dul’kin, J. Zhai, & M. Roth, Phys Stat Sol A (2014). 6. f)

D. G. Choi & S. K. Choi, J. Materials Science 32, 421 (1997). 7. g)

E. K. H. Salje, E. Dul’kin, & M. Roth, Appl. Phys. Lett. 106, 152903 (2015). 8. h)

A. G. Chynoweth, J. Appl. Phys. 30, 280 (1959); Phys. Rev. 110, 1316 (1958). 9. i)

I. J. Mohamad, L.Zammit Mangion, E. F. Lambson & G. A. Saunders, J. Phys. Chem. Sol. 43, 749 (1982). 10. j)

E. Dul’kin, E. Mojaev, M. Roth, Wook Jo, & T. Granzow, Scripta Mater. 60, 251 (2008); E. Dul’kin, B. Mihailova, M. M. Gospodinov & M. Roth, J. Appl. Phys. 115, 084103 (2014). 11. k)

E. Dul’kin, A. Kania & M. Roth, Phys. Stat. Sol. B1, 00145 (2016); Mater. Res. Express 1, 015105 (2014); E. Dul’kin & M. Roth, J. Phys. Cond. Mat. 25, 155901 (2013) 12. l)

E. Dul’kin, J. Zhai & M. Roth, Phys. Stat. Sol. B252, 52111 (2015).

I.1 Pinning

Henrick Barkhausen observed the discontinuous (jerky) nature in the ferromagnetic hysteresis loop in 1919 and used this to explain the presence of domains in the material Rudyak (1971); this obscure behaviour was named after him. This mechanism that leads to the jerky nature of the response to the field. As a field is applied, a unit cell with a magnetisation (polarisation) opposing this field will be flipped when the field is larger than the coercive field, Bc (Ec) Rudyak (1971); Tebble (1955); Yamazaki et al. (2019). This is most likely to occur just beyond the domain wall, as shielding effects are weakest here. This is equivalent to the perspective of the domain wall translating through the material Rudyak (1971); Tebble (1955); Yamazaki et al. (2019); Kagawa et al. (2016a); Salje and Dahmen (2014). As the domain wall approaches a defect, the translation is truncated at this point and the domain wall wraps around the defect – known as pinning. Barkhausen noise has been previously shown to obey a power-law probability distribution Tan et al. (2019); Dahmen and Ben-Zion (2009); Salje and Dahmen (2014), which takes the form:

[TABLE]

where $P[X]$ is the probability, $X_{i}$ is the parameter of the data which obeys a power-law, $X_{min}$ is the lower bound normalisation condition for the power-law – the value of $X_{i}$ that starts to obey the power-law Tan et al. (2019); Salje and Dahmen (2014); Dahmen and Ben-Zion (2009); Clauset et al. (2009). To prevent divergence as $X\rightarrow 0$, normalisation constants are introduced.

There has been a great deal of investigation into Barkhausen noise and comparing the crepitations involved to others that are statistically similar Kramer and Lobkovsky (1996); Sethna et al. (2001); Salje and Dahmen (2014); Dahmen and Ben-Zion (2009); Dahmen and Sethna (1996); Dahmen (2017); Salje et al. (2019). It is surprising how many different systems exhibit a crackling noise that obeys a power-law: crushing a chocolate bar wrapper in your hand Kramer and Lobkovsky (1996) and the famous “Snap, Crackle, Pop!” when pouring milk into a bowl of Rice Krispies™ Sethna et al. (2001); Salje and Dahmen (2014) are some obscure examples. The aim of fitting to a power-law model is to extract the exponent. In a paper by Salje et. al (2019) Salje et al. (2019) several parameters of Barkhausen noise in BTO were fitted to a power-law.

The interesting thing here is the possibility of using this system to investigate earthquakes. It has been previously shown by Dahmen and Ben-Zion (2009) Dahmen and Ben-Zion (2009) that Barkhausen noise and earthquake events share many statistical similarities. Similarly, Salje et. al (2019) fitted the acoustic data for their Barkhausen noise experiments on a BTO single crystal to Omori’s Law, with the exponent p=$1.0\pm 0.2$ Salje et al. (2019).

There have been many different mechanisms/theories proposed for the generation of aftershocks in earthquakes. Sholtz (1968) suggested this is due to the fractures or defects produced by residual stress left after a main fracture Scholz (1968). There are several other explanations follow a similar thought process of relaxation after stress Shaw (1993); Ouillon and Sornette (2005); Wang et al. (2010). In theories such as these, the reduction of aftershock rate is due to the reduction of probability of microfailure; this is due to the reduction of availability of microfailure sites Dyskin and Pasternak (2019). This is an interesting perspective and is applicable to systems in this study; the stress on the material caused by switching of a 90*∘* domain wall could then be relaxed by a subsequent domain switching, leading to an aftershock. Therefore, the diminishing rate is due to the diminishing availability of domains to be switched.

Omori’s law was developed in 1894 following analysis of a magnitude M=8 in Japan in 1891 Utsu et al. (1995); Guglielmi (2017). It states that the frequency of aftershocks that follow the main event decreases with time according to the hyperbolic law:

[TABLE]

where k and c are constants, and t is time Guglielmi (2017); Utsu et al. (1995); Dyskin and Pasternak (2019). It is now more commonly replaced with a more general function Guglielmi (2017); Utsu et al. (1995); Dyskin and Pasternak (2019):

[TABLE]

where p is the power-law exponent and $0.7\leq p\leq 1.5$ Utsu et al. (1995); Dyskin and Pasternak (2019).

The fitting of the Omori power-law to the jerk frequency data from ferroelectric switching experiments alludes to the possibility of designing reproducible experiments on earthquakes using these systems.

I.2 Mean-field Theory

The mean field model simplifies a system with complex behaviour by treating all contributions as a single averaged field. For many physicists, the mean field model is most likely encountered in the study of the many electron problem; where the repulsion on one electron from all the other electrons is approximated to a single potential Bruus and Flensberg (2009). Applying this to slip statistics, the microscopic details of the materials (locations and quantities of defects) are approximated to a mean field Dahmen et al. (2009); Dahmen (2017). The model predicts the critical exponents of slip avalanches with a broad distribution of sizes (energy/amplitude of the jerks) Dahmen (2017). There have been several notable experimental studies that have validated this model Tsekenis et al. (2013); Salje and Dahmen (2014); Salje et al. (2019); Tan et al. (2019); Friedman et al. (2012).

The main assumptions of the model are the following:

• •

The material has weak spots (eg. defects in the material and cracks in the rock for earthquakes) Dahmen et al. (2009); Dahmen (2017).

• •

There is only one tuning parameter (the weakness of a weak site) Dahmen et al. (2009).

As shear stress increases, the weak spots are stuck until the local shear stress exceeds the random threshold of failure for that site Dahmen et al. (2009); Dahmen (2017); this leads to a slip by a random distance. The release in stress from the failure of a weak site is redistributed to other weak points, with some probability of this breaching the failure threshold of the sites that now have an increased shear stress. In this study, the applied field will cause the domain wall to move and this will be the mechanism for shear stress on the weak sites, and the slip distance will be observed as the jerk in current.

The probability distribution of energies of jerks, P(J), is proportional to the energy of the jerk to some critical exponent, $\epsilon$:

[TABLE]

where energy scales as:

[TABLE]

v(t) is proportional to the instantaneous growth rate of the avalanche Dahmen et al. (2009); Dahmen (2017).

The probability distribution is also a function of shear stress. In the experiments in this study, the avalanche energies are taken over the entire stress range. Therefore, it is important to integrate over stress to predict the exponent correctly. This leads to a larger exponent $\epsilon=-\frac{5}{3}$ compared with $\epsilon=-\frac{4}{3}$ when not stress integrated Dahmen et al. (2009); Dahmen (2017); Tsekenis et al. (2013); Salje and Dahmen (2014); Salje et al. (2019); Tan et al. (2019); Friedman et al. (2012).

II Experimental

II.1 Samples

Earlier studies of Barkhausen noise in ferroelectrics Salje et al. (2019); Mai et al. (2015); Salje et al. (2015); Lupascu et al. (2003) emphasised a few well-known materials; a full listing is given in Table 2.

The specimens used in the present experiments were described in earlier papers. Tan et al. (2019); Salje et al. (2019) Three ceramic PZT specimens were examined, emphasising a commercial lead zirconate titanate (referred to as PZT throughout) from PI Ceramic Lederhose, Germany Pie (2019), labelled PIC255 by the manufacturer, and one single-crystal barium titanate sample. In addition BaTiO3 ceramics were prepared that were Fe-doped.

II.2 Electronics

A high voltage amplifier was used to generate pulses up to 1.5 kV in the project but can generate a potential difference of up to 40 kV. The FE-module allows for hysteresis loops to be measured TFM ; this can be replaced by other modules to probe other electronic behaviours of materials TFM . For the high temperature measurements, the sample was placed in a aixACCT piezo sample holder which could be used to set the temperature up to 473 K. For cold temperature measurements, the sample was placed in a closed-cycle cryostat, which could cool down to approximately 80 K by means of the Gifford-McMahon refrigeration cycle Gifford (1966).

II.3 Data

The triangular waveforms generated using the TF-Analyzer software were applied to the sample via the probes and the current and polarisation responses were measured. These waveforms were manually generated using the software’s manual waveform generator. The maximum number of points that can be measured using the software is 1000, hence it was important to choose the recording region to maximise the number of useful data points. The coercive field of ferroelectrics is frequency dependent Scott (1996); Viehland and Chen (2000); the voltage ramp-rate was kept constant throughout the experiments to ensure this was not an issue.

III Avalanche Analysis of Exponents

III.1 Power-law Histogram (PLH)

This is a straightforward method for extracting the exponent. The natural logarithm of the data is taken and the data is divided into bins of equal logarithmic width using the histcounts function in Matlab. This is preferred to binning linearly then taking the natural logarithm due to compilation errors in the code when taking the natural logarithm of an unpopulated bin. The probability was scaled with the width of the bins using equation 6 so that the weighting was correlated to the width.

[TABLE]

where $X_{i}$ is the position at the start of the bin (hence $X_{i+1}$ is the position at the end of the bin), $\tilde{P}(X)$ is the raw population and P(X) is the re-scaled population. The data is denoted X here and elsewhere to acknowledge that this can be several parameters of the system. It is important to re-scale the population, as without this, the linear fit produces an exponent of $\eta+1$ instead of $\eta$ due to an increase in bin width with increasing X. The distribution was normalised using the trapz function in Matlab; this was generally poor as it uses the trapezoidal method Tra (2019). However, this was acceptable in this study as the focus is on the shape of the distribution produced, not the specific values.

III.2 Maximum-Likelihood (ML)

After initial shocks, whether voltage pulses in ferroelectrics or real earthquakes, the frequency of aftershocks subsides. This decrease can be fitted in two ways: (1) It can simply be fitted via least squares to an exponential. This gives a plausible result but it suffers from the fact that it is not obvious where to begin and end the data set being fitted. A more rigorous approach is to use the maximum likelihood procedure, which requires that there be a plateau in certain graphs of the raw data. This procedure is reviewed in refs Tan et al. (2019); Salje et al. (2019); Clauset et al. (2009); Rice (2007); Vassiliev (2017). It is useful in picking out unambiguously where the data do not satisfy a single power law. In such cases the data may be a superposition of two or more power laws, or they may not be power laws at all (e.g., a logarithmic or algebraic decay). In the present study the “waiting time” data between aftershocks do not satisfy this power law requirement.

III.3 Resulting Omori Exponent Aftershocks in PZT

Data for aftershocks in PZT: The Omori Law. Fig. 1 shows the results of 100 runs on PZT at room temperature (298K). The upper curve is for large amplitude voltage pulses (ca. 500-1000 attoJ) and gives an Omori exponent of p = 0.95$\pm$0.03. The traditional value in earthquakes since the 8.0-magnitude Japanese quake of 1894 is 1.0; and in barium titanate at room temperature Salje et al. Salje et al. (2019) found 1.0$\pm$0.2. We emphasise that Salje et al. (2019) measured shocks acoustically whereas we measure them via voltage output; the two signals are not a priori identical (especially in the case of 90-degree domains in pure ferroelastics such as lead phosphate which are not ferroelectric).

III.4 Amplitude Exponent in PZT

Fig. 2 illustrates the fitting of data to the amplitude exponent, $\alpha$ in PZT at room temperature. Here the amplitude is the number of aftershock pulses within a certain bin size. Our value is 2.25$\pm$0.11, in good agreement with the value of 2.23 for barium titanate.Salje et al. (2019)

III.5 Duration Exponent in PZT

Fig. 3 illustrates the duration of aftershocks for PZT at ambient temperature. The data do not exhibit a plateau in the maximum likelihood analysis, and so we conclude that the present data are insufficient to imply a power law dependence or to yield an unambiguous exponent.

III.6 Further Data Versus Temperature for Barkhausen Exponents in PZT

Although we have presented some data Tan et al. (2019) on energy exponents in PZT previously, we extend those here in Table II and Figs. 4 and 5. These show three sets of exponents observed: One near 1.4; one near 1.7, and a third near or above 2.0. In Ref. Salje et al. (2019) three regimes were also observed for BaTiO3, but the shorter middle regime was not attributed to a third, distinct exponent, but instead to a transition region with Poisson statistics. We can only speculate about their physical origins (depinning from point or extended defects).

IV Energy Exponents for BaTiO3 in tetragonal and rhombohedral phases

Fig. 6 illustrates our aftershock data in tetragonal single-crystal barium titanate at room temperature. And Fig. 7 shows data at T=80K. The latter do not exhibit a linear dependence in the log-log graph of pulse height; and in comparison with ambient data, the plateau is much larger and the fall-off more abrupt yet rounded and nonlinear. Neither is there is plateau in the maximum likelihood graph. We conclude from this that the Barkhausen switching dynamics are very different at T=80 K and might not not be characterized by a single exponential dependence. If there is a power-law dependence, it sets in at only high slew rates J. This may be related to the recent observations Mohamad et al. (1982) in lead germanate. Other examples of ferroelectrics with very high coercive field thresholds for Barkhausen noise include Kumari et al. (2016) GaFeO3. See also the results by Y. Tokura’s group Kagawa et al. (2016b) that domain wall creep thermally vanishes in ferroelectrics at low temperatures (replaced by tunneling at the lowest T).

Lowering the temperature to T=80K into the rhombohedral phase gave very different Barkhausen data (Fig. 7). Doping barium titanate with Fe Qiu et al. (2010); Xu et al. (2009) gave too soft a material for Barkhausen data.

V Summary

The present work has resulted in accurate values for the Omori exponent of aftershocks in Barkhausen pulses in PZT, not previously reported. The value obtained is very close to unity, as predicted, and more accurate than previously reported values.

Rather more surprising are the strong temperature dependences for the energy exponents in both PZT and barium titanate; these show as temperature is changed an evolution to two distinct superimposed values, one near 2.0 and one near 1.4. In barium titanate no maximum likelihood exponent can be fitted in its rhombohedral R3m phase at low temperatures (T=80K).

Our suggestion is that in general at least two depinning processes are involved, one involving point defects and the other, extended defects (threading dislocations in PZT). Another possibility [z’] is that one set is Barkhausen noise due to domain walls and the other set due to thermal microcracking.

**Omori data on PZT:

**p = 0.95$\pm$0.03 (large pulse, ca. 500-1000 attoJ); p = 0.94$\pm$0.06 (small pulses, ca. 1-10 attoJ) 2. 2.

Temperature data on PZT:

1. a)

Room temperature (three values for energy exponent): 1.79$\pm$0.08 (lowest slew rate), 1.47$\pm$0.04 (intermediate; this may be a transition regime rather than a separate exponent), 2.04$\pm$0.19 (highest slew rate) 2. b)

Energy exponents at T=160K: 2.10$\pm$0.20 and 1.69$\pm$0.10 3. c)

Energy exponents at T=227K: 1.26$\pm$0.21 and 1.68$\pm$0.18 4. d)

Amplitude exponent: 2.23$\pm$0.11 5. e)

Duration exponent (no unambiguous power law; insufficient data) 3. 3.

**Temperature data on BaTiO3:

**Aftershocks display very different statistics in rhombohedral barium titanate at T=80K, with a long plateau of steady-state followed by abrupt cessation – not a simple power-law dependence according to maximum likelihood analysis. Prabakar et al. (2004) A similar effect was reported some years ago in lead germanate but is not understood. It suggests that the domain motion kinetics in high-coercive field ferroelectrics are limited by nucleation times and not by field-activated creep.

Acknowledgements.

We thank Ekhard Salje, Finlay Morrison, and Jonathan Gardner for helpful discussions. Work supported by EPSRC grant EP/PO24637/01.