Avalanches in ferroelectric, ferroelastic and coelastic materials: phase transition, domain switching and propagation

Abstract

The motion of domain walls and phase fronts in ferroelectric and ferroelastic materials displays discrete, impulsive jumps, i.e. jerks, as indicators of avalanches on a broad range of scales. Several experimental techniques used to characterise these avalanches show that they are power law distributed and share some statistical similarities with the avalanches observed during the compression of porous materials. Simulations attribute their origin to kinks and needle domains pinned by junctions between domain walls and show their evolution with the amplitude of the driving field. Recent studies on conductive and polar domain walls prove promising for the development of domain wall engineering.

KEYWORDS:

• Domain wall
• avalanche
• switching
• universality

1. Introduction

The idea that ferroelectric and ferroelastic domain walls possess their own functional physical properties has been discussed theoretically as early as the 70s [1], followed by seminal experimental works in the 90s [2,3]. More recently, progress in the spatial resolution of atomic force microscopy and high-resolution transmission electron microscopy, and developments in optical techniques [4], have led to the experimental observation at domain walls of unusual conductivity in insulators [5–12] and polar properties in non-polar materials [13–22]. These results opened the way for a new paradigm for devices, coined domain wall engineering, where domain walls rather than domains are the active elements [23–26].

In order to be implemented successfully, domain wall engineering requires the control of domain walls’ movements. Ferroelectric and ferroelastic switching is traditionally described by the nucleation and growth (respectively shrinkage) of domains [27,28] within the phenomenological approach of Ishibashi and Takagi [27]. This approach works for individual movements across long-time scales but not for collective movements across short-time scales, where domain walls exhibit jumps which sometimes trigger other jumps and create avalanches.

Such avalanches are well described within the framework of ‘crackling noise’ [29,30] where the movements of domain walls are characterised statistically by analysing the probability P(J) of a jump J to occur, which is often power law distributed with P(J) ∼ J^-γ F(J) where F(J) is an exponential cut-off and γ a well-defined exponent. Experimentally, a large range of experimental techniques (acoustic emission, depolarisation current measurements, optical microscopy, dynamical mechanical analyser, etc.) have been used to observe the jumps and their distributions. Considerable theoretical work has also been performed in order to provide the tools to analyse these distributions, many of which are based on the maximum-likelihood method [31–33].

Here, we review the physics of jumps and avalanches during ferroelectric and ferroelastic switching processes and phase transitions. We report the exponents corresponding to the power-law distributions of energies, amplitudes and durations of avalanches. We then compare with avalanches observed in coelastic materials under uniaxial compression.

2. Ferroelectric switching and phase transitions

2.1. Avalanches during electric field induced switching

The motion of ferroelectric domain walls under electric field is a nonlinear dynamic process characterised by sudden jerks, giving rise to equivalent jumps – or “noise” – in the measured current. These avalanches are the consequence of the interaction of domain walls with defect centres (which are pinning them), depolarisation fields resulting from local polarisation charges [34–36], and other domain walls [37,38]. If domain walls are organised in a simple pattern, such as stripe domains, they all move simultaneously and give rise to a large single jump. If domain walls motions are correlated but changing patterns (∼ stripes, needles, kinks, etc.) they generate avalanches of greater complexity.

Early studies of ferroelectric avalanches [28, 39–43] were performed by applying the electric field through successive pulses – or step functions – and the observed jumps in current were mainly discussed as an evidence of the existence of a ferroelectric domain structure [44]. The amplitude, duration and number of jumps were used to estimate the volume of material that had changed polarisation. Preliminary analyses of these jumps also provided the first insights into the physics of avalanches in ferroelectrics. It was observed that the number of jumps increases in the steeper section of a hysteresis loop [39,40] and that the number and amplitude of jumps are temperature independent [40]. It was also reported that the number of jumps decreases with increasing jump amplitude [40]. This was a first insight into the power-law distribution of energies reported later. Jumps were found to be positive or negative – inverse polarisation-reversal jumps – and to still occur after the application of the electric field [39,40, 42], suggesting the existence of relaxation mechanisms. Finally, individual pulses were considered to occur independently of each other’s [40]. This conclusion was, however, a consequence of the low spatial and temporal resolution of the measurements at that time, and has been contradicted by recent experiments [45].

Similar approaches were used to study relaxor ferroelectrics and the behaviour of polar nanoregions in Pb(Mg_1/3Nb_2/3)O₃ [46] and 0.88Pb(Mg_1/3Nb_2/3)O₃-0.12PbTiO₃ (PMN-PT) [47]. Under a sinusoidal electric field of maximum amplitude 130 V cm⁻¹, jumps in current were observed in the paraelectric phase of Pb(Mg_1/3Nb_2/3)O₃, where macroscopic ferroelectric domains are absent. From the size of the jumps, it is estimated that switching events involve more than a single polar nanoregion. Complementary measurements were performed under dc-electric field (instead of ac-electric field) and thus in the ordinary ferroelectric regime where jumps in current are induced by the motion of macroscopic domains [47–49].

Each movement of a ferroelectric domain relative to other domains with non-180° polarisation also carries a significant strain component. The domain walls are not only ferroelectric but also ferroelastic. The switching process emits, therefore, elastic waves which can be measured independently of the displacement current. The first extensive acoustic emission measurements were carried out to investigate avalanches in BaTiO₃ under electric-field cycling [45] (Figure 1a). Acoustic emission events arise in this experiment from sudden changes in strain induced by the movement of 90° domain walls (strain at 180° domain walls is too small to induce acoustic emission). The experiment has been performed at a slow rate of 0.5 V s⁻¹ in order to avoid acoustic emission overlapping. The sample has been cycled six times, up to 3 kV cm⁻¹, and the data have been averaged (Figure 1b). The maximum-likelihood method gives an exponent ε = 1.7, extending over four decades (Figure 1c,d). The distribution of amplitudes reveals a slightly higher exponent in the first cycle, which decreases with cycling to τ′ = 2.3 (Table 1). The distribution of waiting times between consecutive events is described by two power-law exponents: for short-time range (1 − Φ) = 0.9 and for longer time range, (2 + ψ) = 2.2.

Figure 1. (a) Experimental arrangement of the acoustic emission measurement in BaTiO₃. The single-crystal plate had a size of 3.22 × 5.41 × 1.00 mm³. The piezoelectric sensor was attached with a thin layer of grease on the large surface while the electric field was applied in an orthogonal direction. (b) Activity plot of the single crystal during six loops. The red curves show the number of jumps while the blue line indicates the equivalent accumulated activity. The jump activities start with a maximum during each loop, then decay and end a loop with another smaller maximum. The sequences are very reproducible except for the first loop of a virgin crystal, which shows higher initial activities. (c) The probability distribution function shows a power-law distribution. (d) The maximum likelihood graph is used to determine the energy exponent. The energy exponent is 1.7 as average over all loops. Reprinted figure with permission from Ref. [45]. ?2019 by the American Physical Society.

Table 1. Exponents describing avalanches in ferroelectric materials resulting from electric-field induced changes in domain structure or phase transitions. ε (energy), τ′ (amplitude), α (duration), p (aftershocks), x (energy-amplitude), χ (amplitude-duration). MF indicates the mean-field (respectively force integrated mean-field) solutions.

Compound	ε	τ′	α	p	x	χ	Refs.
BaTiO3 field switching	1.7	2.3	2.7	1.0	2.0	1.5	[45]
	1.6						[50]
Pb(Zr,Ti)O3 field switching	1.6						[51]
BaTiO3 phase transition	1.3			1.0			[52]
PMN-PT phase transition	1.5			1.0			[52]
MF/force integrated	1.3/1.7	1.7/2.3	2.0/3.0	1.0	2.0/2.0	1.5/1.5	[30]

There is also a clear indication that large events induce aftershocks. Aftershocks correspond to events occurring after an event with an energy within a predefined range (mainshock) and before an event within the same predefined energy range. It is thus possible to compute a local rate corresponding to the number of aftershocks per time interval after a mainshock r_AS. The distribution of r_AS as function of the time distance to the mainshock t_MS exhibits a power-law decay for time intervals below 10 s: $r_{AS}\left(t-t_{MS}\right)={\left(c-t-t_{MS}\right)}^{-p}$ where c is a constant independent of time, and the exponent p = 1 (Omori law). These results indicate that events that occur shortly (<10 s) after a mainshock are strongly correlated and can be classified as aftershocks, while later events are not.

A similar energy exponent has been obtained through simultaneous displacement current and birefringence measurements [50], on the same sample. The measure of birefringence has been performed with an optical microscope working in transmission, which can be used to visualise the domain structure evolution under an applied electric field (Figure 2). For the analysis, birefringence images are divided in several sub-regions where the mean value of all pixel intensities (µ) is computed as a function of time (and hence electric field). As for acoustic emission, µ exhibits sudden jumps, with an amplitude $A=\mathrm{d}\mu /\mathrm{d}t$ and an energy $E\sim A^2.$

Figure 2. (a) Sketch of the experimental setup: polarised white light is transmitted through the 1 × 6 × 1 mm³ BaTiO₃ single crystal (field applied across 1 mm) and an analyser is used to select the polarisation state before the CCD camera. An electric field is applied along the [−110]_c, and displacement current is measured with a picoammeter. (b) The upper panel shows birefringence images for orthogonal angles of the polariser (0° and 90°). The lower panel shows images for the same angle of the polariser but with focus on top and bottom surfaces of the sample, respectively. (c) Evolution of the domain structure during ferroelectric switching at applied voltages of 0 V and 300 V on applying the field, 0 and −300 V on removing the field. In (b) and (c), arrows schematise the direction of the polarisation in the domains. Reproduced from Ref. [50]. CC BY 4.0.

When jumps are extracted from a single region covering the field of view of the optical microscope, the maximum-likelihood method shows a plateau but with large error bars, because of overlap between avalanches (easier to avoid in acoustic emission and displacement current measurements) and poor statistic. When jumps are extracted from a larger number of sub-regions, error bars become smaller, until a critical number of sub-regions (depending on the material and experimental conditions), where most avalanches occur across more than one sub-region, which introduces an energy cut-off and distorts the maximum-likelihood exponent plateau [53].

Error bars are also reduced when the number of images per time is curtailed since the amplitude of the computed jumps becomes larger than the experimental noise. However, if too many images are skipped, too many jumps are ignored, and the statistical significance of the data analysis decreases. This result evidences that working with a high temporal resolution is not necessarily a requirement for the optical image processing. The optimal rate at which the electric field is applied to best observe avalanches is not necessarily the smallest one. Instead, the optimal rate must be chosen such that overlap of avalanches is small (low rates) and jumps large enough to be detected (high rates).

A detailed statistical analysis of jumps in displacement current has been performed during the application of a triangular electric field at 40–60 V s⁻¹ to two commercial Pb(Zr,Ti)O₃ samples [51]. The statistical analysis is performed on the square of the first-time derivative of the current, i.e. the slew-rate squared (J), by analogy with the definition of energy in acoustic emission analyses. The maximum-likelihood method gives an exponent ε = 1.6 (Figure 3).

Figure 3. (a) Probability P(J) versus jumps J for three different temperatures, showing power law mixing near 413 K. (b) Maximum-likelihood graphs at the same temperatures. Reprinted figure with permission from Ref. [51]. ?2019 by the American Physical Society.

When increasing the temperature towards T_c, the maximum-likelihood analysis reveals the power-law mixing effect of two simultaneous exponents: ε = 1.6 (previously described) and ε ∼ 1.8 (Figure 3). This second exponent is tentatively attributed to depinning of domain walls from threading dislocations, known to be high in Pb(Zr,Ti)O₃ (10¹³ dislocations per cm² at least, about one order of magnitude higher than in BaTiO₃ [51]). As the switching is an athermal process, there is no temperature dependence of the avalanche characteristics in this experiment.

2.2. Avalanches across phase transitions

During stepwise ferroelectric phase transitions a phase front spatially separates the high symmetry, paraelectric phase and the low symmetry, ferroelectric phase. The propagation of the phase front is athermal, i.e. thermal fluctuations are not large enough to drive the phase transition [54,55], and requires a change of temperature or the application of an external electric field which modifies the free energy difference between high and low symmetry phases. In such cases, the system jumps from one metastable state to another, and its energetic path is strongly influenced by disorder (dislocations, grain boundaries, chemical composition variations). Thus, a smooth, fully continuous propagation of the phase front rarely exists and instead the kinetics of the transition is characterised by intermittent and discrete avalanches which occur with a certain relaxation time [52, 56], similar to the case of ferroelectric switching.

BaTiO₃ is a prototypical ferroelectric with a first-order tetragonal-cubic phase transition at T_c ∼ 130 °C and an ideal material to study avalanches. At a macroscopic level, the different phases of BaTiO₃ are described by a single-order parameter (essentially the displacement of the Ti from the centre of the oxygen octahedra) [57,58]. At a microscopic level, however, the observed diffuse X-ray scattering [59] indicates disorder in the Ti ions, which occupy one of the eight equivalent off-centre sites along the [111]_pc directions (pc indicates pseudocubic indices).

In the tetragonal phase, a ferroelectric domain structure spontaneously appears [60,61], which is rather well understood in the bulk but complexifies at the surface where the domain wall profiles are modified [62]. At the surface, phase transitions occur at higher temperatures [63–67], as reported in a photoemission electron microscopy study showing that the domain structure (at the surface) persists up to 277 °C and acts as a memory pattern for bulk ferroelectric domains [67]. This observation is in agreement with scanning surface potential microscopy observations [65,66]. A low-energy electron spectroscopy study confirms these results and show that in the vicinity of the bulk phase transition a transient ferroelastic pattern is characterised by fine stripes a few hundred nanometres wide, which are crossing the ferroelectric domain structure at the surface (Figure 4) [63].

Figure 4. Low-energy electron microscopy 20 µm field-of-view image at the phase transition of BaTiO₃ showing the appearance of transient stripe domains (horizontal lines). The vertical stripes correspond to 180° domains. Reprinted from [63], with the permission of AIP Publishing.

In the paraelectric cubic phase, BaTiO₃ exhibits dynamic polar nanoregions [68,69], in a similar fashion as relaxor ferroelectrics. One extremely useful way to study the polar nanoregions is again to measure acoustic emission [70–76]. In particular, it has been shown that acoustic emission events occur at the Burns temperature T_d where dynamical polar nanoregions nucleate and at T* where polar nanoregions become static [70–76].

It is therefore not surprising that acoustic emission has also been successful to investigate the relaxor characteristics of the paraelectric phase in BaTiO₃. Measurements performed on heating and cooling at an average rate of 1–3 °C min⁻¹ reveal that acoustic emission peaks at T_d ∼ 277 °C and T* = 233 °C, like in relaxor ferroelectrics, and at T_c ∼ 131 °C where acoustic emission arises from the propagation of the transition front [77]. Further investigations of the tetragonal–cubic phase transition at a slower heating rate of 0.5 °C min⁻¹ show that acoustic emission events above the noise level are spread between 129 °C and 137 °C, while on cooling at the same rate, signals are concentrated around 129 °C [56]. However, these measurements are based on acoustic emission spectroscopy with moderate time integration. The low activity hinders statistical analyses so far, so that little can be concluded on the exact movement of domains in the coexistence interval. More detailed work is clearly desirable after it has been shown that acoustic emission is a feasible technique to solve this problem.

High-precision thermally stimulated depolarisation current (TSDC) measurements probe directly the first-order parameter, i.e. the time evolution of the spontaneous polarisation. On heating across the tetragonal–cubic transition of BaTiO₃ single crystals previously poled at room temperature, the current peaks (Figure 5a,b) [52]. For fast heating rates (10 °C min⁻¹) a single peak is observed, corresponding to several events overlapping. For slow heating rates (0.05 °C min⁻¹) it is possible to observe individual events, with about 1000 jerks recorded in a 3 °C interval.

Figure 5. Jumps in TSDC during the phase transformation of (a) BaTiO₃ and (c) 0.7Pb(Mg_2/3Nb_1/3)O₃-0.3PbTiO₃ single crystals (001)-oriented, at three different heating rates (10 °C min⁻¹, 0.1 °C min⁻¹, and 0.05 °C min⁻¹). Avalanches are shown as individual jumps with the decreasing heating rate. A single avalanche event in an enlarged scale is shown in (b) for BaTiO₃ and (d) for 0.7Pb(Mg_2/3Nb_1/3)O₃-0.3PbTiO₃. Reprinted from [52], with the permission of AIP Publishing.

The energy of the avalanches is computed as δE = I²ΔtR where I is the maximum value of the current for a single event, Δt the duration of the event and R the resistance of the sample, considered as constant. Log-log plots of the probability densities of energy reveal power-law behaviours for BaTiO₃ single crystals with different orientations. The maximum-likelihood method gives an exponent ε = 1.3 (Table 1), extending over several decades for (001)_pc, (011)_pc and (111)_pc-oriented BaTiO₃ [52].

Avalanches are also temporally correlated. The distribution of waiting times between consecutive events is described by two power-law exponents: for short-time range the exponent (1 – Φ) = 1.1 and for longer time range, the exponent (2 + ψ) = 2.3. The analysis of aftershocks indicates that events that occur shortly (< 10 s) after a main shock are strongly correlated, while later events are not [52].

Similar measurements have been performed across the relaxor ferroelectric phase transition of 0.7Pb(Mg_2/3Nb_1/3)O₃-0.3PbTiO₃ single crystals with three orientations: (001) _pc, (011) _pc and (111) _pc (Figure 5c,d). The observed time correlations are identical to the case of BaTiO₃. However, the energy-distribution reveals a higher exponent ε = 1.5 extending over four decades [52].

2.3. Mean-field solutions

In many cases, simple mean-field models with infinite range interactions have been used to predict the different power-law distributions of avalanches. The ferroelectric material is modelled as a crystal with N sites, each with a local electric dipole moment. Its Hamiltonian depends on the mutual interaction between electric dipoles, the applied electric field (or temperature) and a term accounting for the disorder. Each local dipole moment flips when the local field exceeds the local coercive field. When a dipole flips, it changes the field on other sites, thereby triggering other weak spots to flip the polarisation. The flip process (and the back switching) spreads over the sample and forms an avalanche. With increasing external forcing, further avalanches follow forming together ‘crackling noise’ or ‘switching noise’, reminiscent of Barkhausen noise in magnets.

Switching noise has some stringent characteristics, which have been confirmed in previous work. The first main observation is that jerk observables J are power law distributed with probability distribution functions, PDF, as PDF(J) ∼ J^-γ, where γ is a well-defined exponent. The maximum amplitude A per avalanche is one of the main parameters which scales in switching noise as PDF(A) ∼ A^-τ’. The avalanche energy is the time integral of the squared amplitude over the duration of a single avalanche. This energy is equally power law distributed and scales with an exponent ε = 1.7 in the relaxed (force integrated) mean field scenario [30]. The energy is a typical quantity that is commonly determined in acoustic emission spectroscopy. Alternatively, the energy is sometimes directly measured as the square of the first-time derivative of the displacement current.

The following scaling equality are obtained from mean-field theory [30]: $\tau \mathrm{\text{'}}-1=\left(\epsilon -1\right)x=\left(\alpha -1\right)/\chi .$

Table 1 shows that for the very limited amount of observations, it appears that for the field switching data in BaTiO₃ and Pb(Zr,Ti)O₃ ε = 1.6–1.7 which agrees very well with the theoretical force integrated value of 1.7. Thermal phase transition is BaTiO₃ shows experimentally the expected mean-field values with ε = 1.3. The experimental exponent in 0.7Pb(Mg_2/3Nb_1/3)O₃-0.3PbTiO₃, ε = 1.5, is slightly higher than the mean-field value, which is tentatively attributed to a higher level of disorder due to the relaxor character of the material, leading to a large number of small jumps [52].

3. Ferroelastic/martensitic switching and phase transitions

3.1. Ferroelastic materials

3.1.1. Experimental results

Most of the ferroelectric materials discussed in the previous section are also ferroelastic materials. As such, it is possible to put in motion their ferroelastic domain walls (usually 90° domain walls) by applying a mechanical stress instead of an electric field [78]. The response of the domain walls is then very similar to the response of (non-ferroelectric) ferroelastic domain walls that are discussed in this section.

Broadband acoustic emission has been used to study the thermally-induced phase transition of the ferroelastic Pb₃(PO₄)₂ [56]. However, the number of events above noise was, so far, too small to perform a detailed statistical analysis. In contrast, experiments on LaAlO₃ have been highly successful and comprise a range of complementary scales, ranging from the optical observation of a single needle growing [79–81] to macroscopic measurements of the sample length [82,83], under slowly increasing stress.

The macroscopic measurement has been performed with a Dynamical Mechanical Analyser (DMA) under increasing oscillating external stress in three-point bending and in parallel-plate geometries [82,83]. The overall decrease in length follows an exponential decay, on which height jumps are superimposed. Measurements with stress rates higher than 35 mN min⁻¹ result in only a few jumps while at a slower rate of 3 mN min⁻¹, the number of jumps increases (13,200 in total) and a statistical analysis becomes possible [82]. Interestingly, about 30% of these jumps are positive, i.e. correspond to an increase in length. Within the hypothesis that jumps are caused by pinning/depinning events of domain walls, such positive jumps are in contradiction with the Middleton’s theorem [84] which states that for purely elastic interactions the interface can only move forward in response to the driving force. Backward jumps would only be observed in disordered media and for viscoelastic interfaces [85], which puts emphasis on the influence of the local structural disorder [30]. The distribution of the squared temporal derivatives of the sample height exhibits a power-law behaviour with an exponent ε = 1.6 extending over three decades (Table 2). The same exponent is obtained if positive jumps are removed before the analysis [83].

Table 2. Exponents describing avalanches in ferroelastic materials driven by different fields (compression, strain, stress, electric, temperature). ε (energy), τ′ (amplitude), α (duration), p (aftershocks), x (energy-amplitude), χ (amplitude-duration). MF indicates the mean-field (respectively force integrated mean-field) solutions.

Compound	ε	τ′	α	p	x	χ	Refs.
LaAlO3 compression	1.6						[82,83]
	1.8						[79]
PbZrO3 compression	1.5						[83]
	1.6						[82]
SrTiO3 electric field	1.4						[86]
	1.6						[86]
304L-steel strain	1.8						[87]
Ni-Mn-Ga compression	1.5						[88]
	1.5	3.0			4.0		[89]
		2.0					[90]
Ni-Mn-Ga temperature		2.2					[91]
	1.8	2.6			2.0		[92,93]
	1.7	2.4	4.3		2.0	2.4	[94]
		3.6					[95]
		4.9					[95]
	1.8	2.7			2.1		[96]
+ magnetic field	1.5	2.0			2.0		[92,93]
Cu-Al-Be compression	1.3						[97]
Cu-Al-Be temperature		2.4					[98]
	1.6						[99]
Cu-Al-Mn temperature		3.1					[98]
		2.4					[98]
		2.3					[100]
Cu-Al-Ni temperature		3.1					[98]
		2.4					[98]
		2.2–2.4					[55]
	1.5						[101]
	2.0						[101]
Cu-Zn-Al temperature	2.1						[102]
	2.2			0.2			[102,103]
	1.8						[104]
		3.6	3.5			1.0	[105]
		2.4					[98]
		3.1					[98]
		2.7-2.8					[55]
Cu-Zn-Al strain	2.0	2.7			1.7		[106]
Cu-Zn-Al stress	2.2	3.0			1.7		[106]
Fe-Pd temperature	1.6	2.3			2.0		[107]
	1.6	2.1			1.9		[107]
	2.0	3.0			2.0		[107]
MF/force integrated	1.3/1.7	1.7/2.3	2.0/3.0	1.0	2.0/2.0	1.5/1.5	[30]

The distribution of waiting times between consecutive events is described by a single power-law exponent (2 + ψ) = 2.0 for long-time range. The distribution for short-time range could not be analysed because of the low sampling rate of the experiment [82].

It is interesting to compare the macroscopic measurement with the optical observation of the motion under external stress of a single domain in LaAlO₃ in order to assess which propagating part is generating avalanches. The observation of a needle [79–81] shows that the sides remain smooth during propagation and do not exhibit jumps. Thus, they are not at the origin of the avalanches. On the contrary, the front line of the needle tip shows meanders and its progression follows an exponential relaxation envelope, punctuated by jumps (Figure 6). Tracking the movement of the needle tip x(t) yields the kinetic energy E² = (dx/dt)², whose distribution gives an exponent ε = 1.8 [79]. This exponent is related to pinning near surfaces and is slightly higher than the overall exponent 1.6 of needle movements in the bulk of the material [82,83].

Figure 6. Time evolution of a wedge shaped domain after (a) attachment to the surface, (b, c) partial attachment, and (d) after the link snapped. Reprinted from [80], with the permission of AIP Publishing.

Macroscopic DMA measurements in a parallel-plate geometry have also been performed as a function of temperature [82] on a single crystal of PbZrO₃, which undergoes a phase transition from a paraelectric Pm$\overline{3}$m to an antiferroelectric orthorhombic Pbam phase at T_c = 503 K. At room temperature (295 K), 3800 jumps are recorded, 10% of which are positive. The distribution of the squared temporal derivatives of the sample height exhibits a power-law behaviour with an exponent ε = 1.6 extending over three decades. Note that in a similar measurement ε = 1.5 was reported [83], but the typical error margin for DMA experiments on the exponent is ±0.15. At 323 K and 373 K, the same exponent is found. However, at 463 K the behaviour differs considerably, resulting in an exponential distribution, instead of a power-law distribution. This difference has been attributed to thermal fluctuations, which decrease the energy required for depinning domain walls and thus the number of large jumps, in agreement with simulations described in section 3.1.2. At 373 K, the distribution of waiting times between consecutive events is described by a single power-law exponent (2 + ψ) ∼ 2.0 for the long-time range. At 463 K, the exponent becomes 2.9, indicating that long waiting times are suppressed.

More recently, electric-field induced avalanches from domain walls [86] have been reported in the (non-ferroelectric) ferroelastic SrTiO₃. Ferroelastic domains appear during the transition from the cubic to tetragonal phase at 105 K and self-organize in two types of ferroelastic twin domains: {a, c} when the projection of domain walls on the plane (001) are parallel to the [100]_pc or [010]_pc axes and {a₁, a₂} for domain walls parallel to [110]_pc or [1-10]_pc. The ability to move ferroelastic domain walls with an electric field, which is unusual for a non-ferroelectric material, is largely a consequence of the higher polarizability of ferroelastic domains in the plane normal to the tetragonal axis [108]. The motion of domain walls was observed with an optical microscope on a (001)-oriented single crystal: in reflection the {a, c} twin domains are visible because of the corrugation they create in the sample surface, in transmission the {a₁, a₂} twins appear because of the difference in birefringence (Figure 7a).

Figure 7. (a) Birefringence image at 0 and 200 V mm⁻¹. The arrows indicate the {a₁, a₂} twins which double their period at 200 V mm⁻¹. (b) Maximum-likelihood exponent, MLE, (black error bars) showing a plateau expanding along two decades indicated by the red dashed line. (c) Log-log plot for the jerk spectra distribution (black line). The best fit at the same energy range as the maximum-likelihood plateau corresponds to a slope of −1.6 (red dashed line). Reproduced from Ref. [86]. CC BY 4.0.

The analysis is the same as described in section 2.1 for BaTiO₃. For a simple pattern of stripe {a, c} twins, the maximum-likelihood method gives an exponent ε = 1.4 indicated by the onset of a plateau. For a simple pattern of stripe {a₁, a₂} twins the maximum-likelihood method gives an exponent ε = 1.6 extending over two decades (Figure 7b,c). This is the only experimental observation, among ferroelastic and ferroelectric materials, of an exponent depending on the type of twins/domain walls. For a complex pattern of domains, characterised by an intricate superposition of {a, c} twins in two orthogonal directions, the motion of domain walls is smoother, with fewer jumps. The maximum-likelihood method reveals an exponentially damped power law with ε = 1.6. This damping is attributed to the pinning of domain walls by other domain walls [86].

These avalanches are observed at 6 K, where domain walls move strongly under electric fields, but they disappear above 40 K where the domain structure is only weakly sensitive to the electric field. Thus, the maximum-likelihood exponent above 40 K does not exhibit a plateau or deflection, and is similar to the behaviour expected for experimental noise [86].

3.1.2. Simulations

The experimental observation of different energy distributions in SrTiO₃ depending on the type of domain walls and domain patterns reveals the complexity of avalanches and the need of simulations to understand the collective motion of domain walls under external stress or temperature. We focus here on large stresses leading to irreversible changes of the domain pattern, rather than on small mechanical excitations like in resonant ultrasound spectroscopy [109].

Complex domain patterns are well reproduced with molecular dynamics simulations of a simple toy model [110–113]. This model is based on a two-dimensional square lattice with a shear angle fixed to 4°, which is like the value for typical ferroelastics (Figure 8). (When extended to a three-dimensional domain pattern, the model gives results similar to what is described below [114].) Three interatomic interactions are considered: harmonic springs along the sides of the sheared unit cell (nearest neighbours), Landau springs in the diagonal (next nearest neighbours) and fourth-order springs between the third nearest neighbours. Free boundary conditions are adopted in order to allow for the nucleation of needle domains at the surface. Note also that atomic scale simulations are preferred to force-field simulations because the propagation of needle domains and kinks are well reproduced with the former but averaged over with the latter [110].

Figure 8. Spring model with two charged sub-lattices. Coulomb interactions and interatomic interactions are combined in the model. The interatomic interactions between the nearest neighbours are harmonic (shown by springs). Nonconvex interactions (gray sticks) along diagonals in the ferroelastic sub-lattice lead to the formation of domain structures. The electrostatic interaction between the two sub-lattices combines with weak 2nd or 6th order springs are used to mimic the repulsive coupling between the sub-lattices. The spring stiffness for 2nd and 6th order potentials is 0.1 and 600, respectively. Reprinted from [113], with the permission of AIP Publishing.

The initial condition contains only one domain wall. Under shear strain this domain wall moves sideways and needle domains nucleate at the surface and grow. On further increasing the stress the number of needle domains increases even more. Some needles grow orthogonal to the first ones, leading to junctions between domain walls. Several kinks also appear in the domain walls. One of the main results of this simulation is that the intersections between domain walls act as pinning centres for kinks and needle domains and are at the origin of jumps. The energy distribution of these jumps follows a power-law behaviour with an exponent ε ∼ 2. Thus, external defects are not required to generate avalanches, intersections between domain walls alone are enough [110].

A more detailed simulation shows that the structure of the avalanches and the amplitude of the jumps are different in the elastic regime, in the vicinity of the yield point and in the plastic regime of the material [114]. The amplitude of the jumps is higher in the yield regime – where most of the ferroelastic domains are formed – compared with the amplitude of the jumps in the elastic regime – where some nucleation of kinks happens – and the plastic regime – where de-twinning occurs. Near the yield point, small jumps superimpose on top of large jumps. They correspond to sub-avalanches that follow each other without the main avalanche coming to rest. The value of the energy exponent is ε ∼ 2 in the elastic regime, ε ∼ 1.7 close the yield point and ε ∼ 1.5 in the plastic regime (at 1 K). The distribution of waiting times between consecutive events is described by a single power-law exponent (2 + ψ) = 2.0 (at 1 K).

Jumps are typically temperature dependent and correlate with the evolution of the domain structure. At low temperatures, only a few jumps are observed, and the domain structure is rather simple (Figure 9a). With increasing temperature, the domain structures complexifies (Figure 9b–d), the number of jumps raises, and the energies follow an athermal power-law distribution. ε decreases with increasing temperature, and thus with increasing density of the ferroelastic pattern: from 1.9 (0.5 K), 1.7 (1 K) to 1.3 at higher temperatures (10 and 20 K) [114]. At even higher temperature, de-twinning occurs (Figure 9e,f), thermal fluctuations generate many jumps, the power-law behaviour is lost and replaced by a thermally activated behaviour with constant activation energies, typically a Vogel–Fulcher process [110, 114].

Figure 9. The temperature effect on domain nucleation near the yield point for six different temperatures. (a) No complex domain pattern occurs at very low temperatures. With increasing temperature, the complexity of the domain pattern increases from (b) to (d) and decreases from (d) to (f). The colour scheme represents the local shear angle from the underlying bulk structure The most complex patterns (highest number of junctions) occur at 1 and 10 K. Reproduced from Ref. [114]. CC BY 4.0.

Interestingly, nucleation and propagation of kinks lead also to the emission of acoustic phonons, which exhibit avalanches over a large temperature interval near the Vogel–Fulcher point [115]. They are characterised by energy exponents ranging from 2.5 to 3.0, higher than the energy exponents of kinks, needle domains and their pinning by junctions discussed above. There is also a clear indication for aftershocks: the number of aftershocks per time interval after a mainshock exhibits a power-law behaviour with an exponent p = 1.0 for time distances below ∼1 ps. The distribution of waiting time between consecutive events is described by a power-law exponent for short-time range (1 – Φ) ∼ 0.7, but no scaling behaviour is observed for longer time range.

While it is clear from simulations that movements of kinks, needle domains and their pinning by junctions lead to avalanches, it is reasonable to ask if these are the avalanches observed experimentally. For example, optical microscopy does not have the spatial resolution to probe the movement of individual kinks and is instead probing avalanches resulting from a combination of many of these movements. The same applies to DMA measurements where the resolution in sample height is about 3 nm [82]. The answer is less straightforward for acoustic emission measurements. Simulations [116], based on the previously described toy model, have been performed to analyse the movement of surface atoms which would transmit the acoustic wave to a piezoelectric transducer, as in a real acoustic emission experiment. They show that, at the yield point, the nucleation of needle domains and the collapse of domains extending across the thickness of the sample, involves a large energy drop at the surface of 3.7 meV/atom and 0.56 meV/atom, respectively, easily detected with acoustic emission [88]. In the plastic regime, movement of needle domains and kinks involve energy changes at the surface of 0.017 meV/atom, which could also be picked-up by acoustic emission measurements [88]. Simulations are therefore very useful to understand the origin of the avalanches measured in ferroelastic materials. They can also give key indications to generate experimentally complex ferroelastic patterns and make use of avalanches in applications.

The most common route to produce high domain wall densities is by fast temperature quench from a high-temperature paraelastic phase into a ferroelastic phase [117]. Simulations show another method which consists in applying shear strain in the low-temperature ferroelastic phase [118]. This is done by sputtering a ferroelastic thin film on a bulk ferroelastic substrate, at high temperature. Upon cooling, the substrate lattice parameters decrease and impose a shear strain on the film. Once the film reaches the yield point, a high density of ferroelastic domains spontaneously appears, generating a high density of junctions. This pattern remains essentially stable under further strain. It is found that soft thin films (nearest neighbour spring constant k₁ = 10 in the model) have higher densities of domain walls than hard thin films (k₁ = 20), such that the domain walls occupy ∼5% of the total volume of the sample [118].

In term of applications, these complex domain patterns are very appealing: their formation absorbs most of the energy when a sample is hit by high-speed objects (such as in bullet proof vests) [119] and their kinks can move at supersonic speed and be used in devices operating at GHz frequencies [120]. In specific geometrical configurations (such as rough surfaces), it was found that ferroelectric hysteresis behaviour and piezoelectricity can be mimicked by purely ferroelastic materials if the domain walls are polar [121–123]. These results were obtained with molecular dynamics simulations of a charged model (Figure 8) [113], revealing polar vortex structures [124] and a net macroscopic polarisation even in materials which are non-polar by symmetry [125] (SrTiO₃ for example).

3.2. Martensitic materials

In martensitic materials, avalanches have been investigated with acoustic emission, length variation measurements, calorimetry or direct optical observations of the microstructure. Acoustic emission and jumps in calorimetric data arise from the dissipation of elastic energy associated with differences in the lattice parameters at the phase front between martensite and austenite phases or from irreversible dissipative processes such as dislocation motion. Length variations result from the motion of ferroelastic domain walls that can be directly monitored optically. Some of the exponents reported vary for the same system, depending on e.g. the rate at which the driving field is applied or the microstructure of the sample, as discussed in several reviews on avalanches in martensitic materials [30, 92, 98, 126]. Here we only report the exponent values obtained (Table 2) and we refer the reader to the original articles for details.

4. Coelastic materials

Many materials which might have been expected to be ferroelastic from symmetry arguments turn out not to be switchable because, for example, their coercive stress is higher than their failure point [127]. In another scenario, switching is not allowed by symmetry, but the spontaneous strain of the transition still dominates the transition mechanism. These materials exhibit a large elastic deformation and/or an elastic anomaly and thus are named coelastic materials [127]. This is the case of quartz, SiO₂, which undergoes a β–α transition at 847 K with a significant elastic precursor softening [128] and a large spontaneous strain generated by tilts of deforming tetrahedra [129].

Avalanches have been reported during uniaxial compression of SiO₂-based porous materials. Contrary to ferroelastic materials where they result from the propagation of domain walls, they are induced by the collapsing of pores and cavities, making a comparison between both systems desirable [83]. In a first set of experiments, acoustic emission has been recorded during the compression of three synthetic glass (Gelsil, Vycor) and three natural sandstones [130]. (Note that the data on Vycor were in fact acquired previously [131,132].) Samples were placed between two aluminium plates, one static, and one pulled downward at constant stress rate. Piezoelectric transducers embedded in the plates, centred 4 mm away from the sample surface, detected the acoustic emission signal.

The maximum-likelihood method gives an exponent ε = 1.4 extending over four decades for Gelsil and five decades for Vycor (Figure 10, Table 3). In sandstone, ε = 1.5 but with a small deviation from the power-law behaviour, indicated by a slope in the maximum-likelihood plot (Figure 10). It is suggested that when there is a strong adherence between grains (Gelsil) or no grains at all (Vycor), the dominating mechanism during failure is the breaking of bonds, which gives a power-law behaviour. In the case of sandstones, SiO₂ grains are cemented together by other minerals, the adherence between grains is reduced, the friction higher and the power-law behaviour distorted. ε is closer to the mean-field value than to the force-integrated mean-field value, which suggests that the failure of SiO₂-based porous materials evolves through a series of critical points, rather than a single critical point at the complete breakdown of the sample. This hypothesis is consistent with the non-stationarity of the signals (acoustic emission and height variations) [130].

Figure 10. (a) Log-log plot of the energy distribution of the acoustic emission events. From top to bottom curves correspond to the following samples, respectively: Vycor, Gelsil (Gel5, Gel2.6), sandstone (LGsan, Rsan, and Ysan). Details of the samples are given in Ref. [130]. For the sake of clarity, except for Vycor, the curves are shifted. (b) Corresponding fitted energy exponents as a function of the lower fitting cut-off recorded for the same samples. The symbols correspond to Vycor (solid circles), Gel5 (squares), Gel2.6 (empty circles), LGsan (empty up triangle), Rsan (solid up triangle), and Ysan (down triangle). Only few representative error bars are shown. Reprinted figure with permission from Ref. [130]. ©2019 by the American Physical Society.

Table 3. Exponents describing avalanches in coelastic materials driven by uniaxial compression. ε (energy), τ′ (amplitude), α (duration), p (aftershocks), x (energy-amplitude), χ (amplitude-duration). MF indicates the mean-field (respectively force integrated mean-field) solutions.

Compound	ε	τ′	α	p	x	χ	Refs.
Gelsil compression	1.4		2.6	0.7–0.8			[130]
	1.6						[83]
Sandstone compression	1.5		3.0	0.7–0.8			[130]
	1.5						[83]
Vycor compression	1.4		2.0	0.8			[130]
	1.5						[83, 133]
MF/force integrated	1.3/1.7	1.7/2.3	2.0/3.0	1.0	2.0/2.0	1.5/1.5	[30]

This behaviour is different from that of typical ferroelastic materials under uniaxial compression where ε reaches 1.8 (Table 2) and seems to indicate that the yield point where most of the jumps occur in typical ferroelastics is a critical point. The effect of dynamics during creep experiments was discussed in Ref. [134] which may explain why criticality is absent in coelastic Vycor.

The determination of an exponent for the distribution of durations is challenging with acoustic emission because of the limited response of the transducers and the use of a threshold in amplitude. Still, the maximum-likelihood method gives α = 2.0 for Vycor, 2.6 for Gelsil and 3.0 for sandstone, indicating a higher number of long duration events in Vycor. The distribution of waiting times between consecutive events is described by two power-law exponents: for short-time range (1 − Φ) ∼ 1 and for longer time range, (2 + ψ) ∼ 2.5 [130].

In a second set of experiments, the same SiO₂-based porous materials have been investigated with a DMA [83]. The measurements were performed in a parallel-plate geometry and several stress cycles were repeated in order to increase the statistics. The maximum-likelihood method gives ε = 1.5 for Vycor, 1.6 for Gelsil and 1.5 for sandstone, slightly different for Vycor and Gelsil compared with the exponents obtained from acoustic emission [130]. This might result from a fundamental difference between the definition of energies in both measurements: in acoustic emission the energy is obtained by integrating the amplitude over the duration of the event, while in DMA measurements at very low frequencies (near the adiabatic/isothermal crossover point) the energy is defined as the square of the first-time derivative of the height [83].

On a local scale, dynamic nanoindentation measurements has been used to probe the hardness, elastic modulus and material’s resistance to penetration, as a function of the indentation depth [133]. This was done with a diamond Berkovich tip with a radius of about 100 nm driven at 100 Hz. The energy is computed as the square of the first derivative of these parameters’ depth dependences [135]. The maximum-likelihood method gives ε = 1.5. The best likelihood exponent plateau is obtained for the elastic modulus, which directly mirrors the breaking of bonds in the sample and therefore confirms the mechanistic interpretation discussed previously for the interpretation of the acoustic emission measurements [130].

5. Conclusions

In this review, we discussed how the dynamics of complex ferroelectric and ferroelastic domain patterns under external fields can be described by power-law statistics at low temperatures and Vogel–Fulcher statistics at higher temperatures when thermal fluctuations dominate. Simulations show that the experimentally observed avalanches find their origin in kinks and needle domains pinned by external defects or/and junctions between domain walls. We also highlighted the statistical similarities between avalanches in ferroelectric/ferroelastic materials and avalanches induced by pores collapsing in coelastic materials.

Within the framework of domain boundary engineering, the next step is to focus on avalanches from domain walls possessing their own functional properties, as recently done for BaTiO₃ [50] and SrTiO₃ [86,125].

Acknowledgments

G.F.N. thanks the Royal Commission for the Exhibition of 1851 for the award of a Research Fellowship. E.K.H.S. is grateful to EPSRC for support (grant EP/P024904/1).

ORCID

Guillaume F. Nataf http://orcid.org/0000-0001-9215-4717.