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Towards Resolving Landauer's Paradox Through Direct Observation of Multiscale Ferroelastic-Ferroelectric Interplay

Electric-polarization reversibility in nano-ferroelectric structures renders them as a convenient platform for exploring phase transitions and developing energy-efficient switching devices. However, the fundamental question of how ferroic domains switch, i.e. how the polarization changes from one state to another, is yet to be answered fully. There are contradicting models and a wide body of accumulated data which disagree as to whether the switching requires domain nucleation. Moreover, ferroelectric domains switch under electric fields that are supposedly too weak to form nucleation sites, indicating that the level of disorder seen in real systems plays an important role. This longstanding so-called Landauer's paradox is the ferroelectric equivalent to the absence of raindrop formation in a dust-free vacuum, leading to supersaturated vapors that cannot exist otherwise, e.g. in spinodal decompositions or inhomogeneous nucleation environments. Here we show that polarization switching in ferroelectric-ferroelastic systems comprises domain types that differ by symmetry, lengthscale and switching energy. These domains switch simultaneously thanks to intermediate-range order of organized pinning sites, supporting the previously-unexplained coexistence of nucleation-and-growth and nucleation-frustrated mechanisms. Our treatment is applicable to other Kolmogorov-Avrami systems with multi-scale phase transitions. Finally, we demonstrate augmented electromechanical coupling based on the collective motion of pinning sites, which is promising for nano electro-mechanical and low-power switching devices.

There is extensive recent literature on self-patterning and faceting and the resulting geometry of domain walls, especially for thin-film ferroelectrics and other two-dimensional systems. This high level of interest is driven by the fact that, as Berge et al. have emphasized [1], faceting is not permitted at thermal equilibrium in two dimensions. Because the perimeter of a two-dimensional structure is one-dimensional, thermodynamically it cannot exhibit long-range order at finite temperatures [2,3]. However, two-dimensional faceting can occur dynamically during domain switching processes and has been modelled numerically for growth previously [4]. In the present work we take a fresh look at domain geometries in ferroelectric thin films that are repetitively switched. We build on earlier work by Du and Chen [5] (expanded by Tagantsev [6] and Hesse, and Alexe [7]), Kalinin [8], Gruverman [9], Waser, [10] and Salje [11], emphasizing comparisons with Kolmogorov-Avrami-Ishibashi (KAI) [12] switching (controlled by domain wall mobility) and nucleation-limited switching (NLS) models [5,6] in non-uniaxial ferroelectric crystals. Our results show that ferroelectric domain switching is a multiscale process with a strong dependence on the ferroelastic-ferroelectric interplay.
We focus in particular on biferroic systems, where the characteristic length scales for polarization correlation (ferroelectricity) and strain (ferroelasticity) differ significantly, being typically of the order 1 micron and 10 nm, respectively. This leads to a hierarchy of mesoscopic structures, some of which exhibit pseudo symmetry, i.e. domain symmetries higher than those of the underlying crystallographic lattice. Such mesoscopic block arrays (e.g. polytwins) can qualitatively change the switching dynamics from the atomic-scale creep of domain walls in lowdefect systems (observed first by Little [13]) to array-jumping, reminiscent of Barkhausen jumps in magnetism. Tybell et al. showed [14] how domain-wall creep exponents depend upon defects in ferroelectrics, but did not consider jumps of entire blocks. A convenient framework to discuss the array jump is "bundle" domains [15], where the ferroelastic polytwins help define a mesoscopic-scale ferroelectric behavior. However, other polytwin systems typical to ferroelectricferroelastic systems also exhibit similar collective behavior. An analogous bundle-jumping is the well-known 1962 Anderson model of Abrikosov vortex motion in Type-II superconductors [16,17]. In each case, an unknown (but repulsive) intra-site force is involved. In addition, bearing in mind the highly-anisotropic one-dimensional nature of ferroic domain walls, low-mobility ferroic domain switching may follow de Gennes' "repetition" model in polymers, in which onedimensional chains slide like spaghetti through each other. Similar low-wall mobility has recently been attributed to observations of oxygen vacancy footprints [18], which in turn can lead to unconventional electric and mechanical coupling with magnetism [19].
The essence of ferroelectricity is the reversibility of polarization under an external electric field. Therefore, the most important and fundamental question to answer in ferroics is: is there a single model to describe the domain switching? Having such a model is also technologically significant, e.g. for commonly-used low-power switching devices [20]. Traditionally, two competing mechanisms have been suggested to explain this switching. The first is the KAI model, in which domain switching is equivalent to a phase transition in an infinite space. In this model, application of an external electric field gives rise to randomly distributed nucleation points. Upon increasing the electric field, these nuclei merge to form a single domain with common macroscopic polarization. Quantitatively, the fraction of the switched area satisfies: where t0 is the switching time and n is the dimension (n = 2 for thin films). The KAI model assumes a constant nucleation growth rate. However, experimental results, especially those involving high electric fields, challenge this assumption [6,7]. Hence, an alternative NLS approach was also suggested. Here, the domain switching occurs when the polarization in many uncorrelated elementary regions is reversed independently [5,6] and the fraction of the switched area satisfies: where 1/i is the nucleation rate within an elementary region i, i is the area of region i normalised by the average area of elementary regions in the ferroelectric and <…>i is the average over the different regions i.
Gruverman et al. observed [9] polarization switching in lead zirconate titanate (PZT) directly, by means of piezoresponse force microscopy (PFM [21][22][23]). Specifically, they measured the evolution of an area under gradually-switched polarization as well as the number of nucleation sites as a function of switching time. Their data shows that the switching involves two distinguishable steps and they were able to quantitatively fit the first step to the KAI model and the second step to the NLS model. Hence, a decisive conclusion about which of the switching mechanisms governs the polarization reversal has remained elusive, as the two models are incompatible. We offer here an alternative interpretation to these results. We suggest that indeed, there are several independent switching processes. However, these processes take place simultaneously, albeit at different length-scales. Specifically, by introducing a multiscale ferroelectric-ferroelastic interplay, we explain the complex polarization reversal mechanism in ferroelectrics that is consistent with the presumably contradicting existing models and experimental results. The relationship between length-scale and switching time have been proposed for various cases, including for nucleation and growth [24][25][26]. However, here we suggest a more general relationship between the two parameters. We note that although most ferroelectrics also exhibit ferroelasticity due to a change in the crystal class during the paraelectricferroelectric transition, some uniaxial ferroelectrics do not [27], and hence require a more subtle treatment [28].
Most ferroelectrics, especially in thin-film form are composed of different crystallographic domains. Such domains arise to compensate for stress accumulated during the ferroelectric phase transition due to, e.g., substrate clamping. As a result, out-of-plane polarization switching involves both electric and electro-mechanical interactions, and thus, at least two different switching mechanisms. We have recently demonstrated that mesoscopic domains with ordered arrays of crystallographic domains should be treated as individual ferroelectric domain entities [15], as exemplified in Figure 1. Such mesoscopic structures can include polytwins or mash ferroelasticferroelectric platform. Here we focus on bundle domains. It has been shown that such bundle domains can switch as a whole, producing a net out-of-plane mesoscopic polarization reversal [26,29,30]. Generally, the bundle-domain size is comparable to or larger than the film thickness [31], but is smaller than the electrodes used to apply the electric field (this is valid for all ferroelectric-based devices and for most relevant experimental studies of domain switching kinetics in thin films). When discussing domain switching, one should take into account both the out-of-plane motion of the atomic dipoles (180° switching) and the reorientation of the ferroelastic twinning planes associated with bundle switching. Although these two switching mechanisms are different in nature [32,33], the switching of mesoscopic/macroscopic out-of-plane polarization involves both of them. force microscopy (AFM) imaging] shows the grain distribution, as well as some of the striped ferroelastic domains (the bundle domain boundary is discussed in more detail in Ref. [15]).
The existence of such a dual mechanism that involves both 180° and ferroelastic domain switching is supported e.g. by the unexplained two-step switching mechanism observed by Gruverman et al. [9] and more recently by Khan et al. [33]. The creep description of the domain kinetics is valid for fast and continuous motions [14,34] Given that Eq. 2 can be approximated by Eq. 3 easily, the presence of multiple pinning sites gives rise to the same mathematical expression that describes the NLS. In the creep framework, the rate of change of the switched area with time, dp/dt is the relevant parameter rather than p(t)).
Hence, by taking the time derivative of the data from Gruverman et al. [9] an obvious separation between the two processes is distinguished on a dp/dt vs. t graph (Fig. 2). The domain-wall length is generally much larger than the extent of individual pinning sites, so that the average influence of the latter is negligible [6]. Nevertheless, in bundle domains, the pinning sites are ordered (meeting points between three or more crystallographic orientations), and they significantly influence the domain stability and mobility [35]. Hence, we can conclude that the switching process in ferroelectric-ferroelastic systems, such as PZT, takes place simultaneously over two primary length-scales. The first is a local fast 180° domain switching, described by the Likewise, we would like to suggest that the KAI switching is associated with Coble creep of a symmetrically-growing domain due to diffusion across the domain boundary.. A difficulty with the KAI model is that experimentally, an increase of the applied field does not change the shape of p(t) as predicted by this model, rather, it shifts the curve along the time axis [6,9]. We would like to propose that this shift results from a delay in the switching initiation. Following Harrison and Salje [11], the initiation of domain switching occurs as an avalanche, with a probability depending on the applied field (recent studies suggest a different behavior when non-180⁰ domains are absent [39]). We found this model in agreement with recent experimental studies that ferroelastic and ferroelectric domain switching occur at different voltages [33]. Hence, the shift in switching initiation with applied field represents the change in probability to avalanche, which is also mediated by ferroelastic domain switching.
To verify the two-step domain switching mechanism by means of direct observation, one has to demonstrate that (i) both local 180° and mesoscopic bundle-domain switching can occur simultaneously in the ferroic film; and (ii) the energy associated with the two switching processes is different (following Jesse et al. [8]). We examined the influence of ordered pinning sites on the domain motion in PZT films [40] via enhanced piezoresponse force microscopy (EPFM) [41]. In EPFM, the polarization of ferroelectric domains is imaged in phase-mode (areas with in-plane polarization appear with a lower contrast) and the ferroelastic domains (non-180°) are imaged in amplitude-mode (ferroelectric domain walls appear as closed dark lines). The topography is also simultaneously mapped in AFM mode, in which the periodic ferroelastic domains are occasionally distinguishable. Because the domains are imaged at nanoscale resolution [41], and no top electrode short-circuits the domains, domains of different lengthscale are imaged and switched independently. Figure 3 illustrates that the proposed KAI-like step is possible, i.e. out-of-plane polarization is switched locally within a bundle domain without moving twinning planes. We first used amplitude and phase EPFM modes for imaging the native striped ferroelastic-domain (Fig.   1a) and out-of-plane polarization distribution (Fig. 1b) in a grain that is much larger than the 60 nm film thickness (and hence is considered as a single crystallite). The crystallite contained two main bundle domains, while the out-of-plane polarization (Fig.1b) divided the grain into two type was discussed elsewhere [31]).

Figure 3| Effect of local polarization switching on the ferroelastic-ferroelectric interplay. (a)
The ferroelastic domain distribution (imaged with EPFM amplitude mode) in the crystallite of interest was not affected by the local polarization switching at chose points within the bundle domains after LHLMs were measured at the points designated by green dots and the point "A" in When the same excitation is applied to a bundle domain boundary, the reversal in the out-ofplane polarization is accompanied by ferroelastic domain switching, which in turn is associated with a specific switching energy. An external field was then applied while positioning the tip at the bundle-domain boundary, i.e. exciting the domains exactly at the pinning sites, where the domain switching is prone to nucleate (point B in Fig. 1b) [26]. Indeed, the external excitation this time moved the ferroelastic domains, i.e. the bundle domain wall (Fig. 4). LHLMs allow one to compare the energy associated with domain switching. Although there have been attempts to attribute energetic differences to simultaneous motion of ferroelectric and ferroelastic domains, typically, this comparison has not been done in conjunction with the ferroelectric and ferroelastic domain switching imaging [42,43]. However, by comparing the EPFM images of the native domain distribution and the redistribution as a result of the LHLMs, we were able to identify the energy profile with a specific switching process. The electromechanical energy (the area within the hysteresis curve) associated with the two domain-switching mechanisms--bundle-domain switching (Fig. 5a), and 180° domain switching only (Fig. 5b)--is compared. The recorded in-phase signal LHLMs are given in Fig. SI1. A clear difference between the energy of these two processes is observed. Our direct characterisation of hysteresis behavior as a result of the different domain motion also supports previous suggestions based on indirect measurements of electromechanical augmentation due to non-180° domain motion by, e.g.
The imaging enhancement of EPFM requires the usage of soft cantilevers (<< 10 N/m) [41] Using such soft cantilevers reduces the confidence in quantitative analysis of the piezoresponse signal in terms of absolute values, but does not affect the confidence in comparative measurements [46,47]. To complete the quantitative analysis, we also performed a macroscopic hysteresis loop measurement, which is complementary to the LHLMs (Fig. 5). Here, we swept the electric field in a plate capacitor that includes the substrate of the sample and a 1 mm diameter top Pt electrode.
The plate capacitor was connected to a 198 nf capacitor in a Sawyer-Tower scheme, so that the polarization (P) was measured as a function of the applied electric field (E) as shown in Fig. 5c. is more than twice as big as that in (b), suggesting that the electromechanical energy associated with the switching is around twice as big -perhaps the ability to rotate domains in-plane will be useful for energy harvesting applications. In-phase LHLMs are given in Fig. SI1. 4 indicates that the free energy of our system at the macroscopic scale is given by: G(P) ≈ -1.7 . 10 7 P 2 + 2.5 . 10 8 P 4 , where P is given in V/m. Although the use of soft cantilevers and the consequent complex factors contributing to the cantilever deflection do not permit a similar straightforward derivation at the meso-and nano-scale Ps is larger for the case of non-180° switching (Fig. 5a). Bearing in mind Eq. 4, this suggests that the potential barrier to be overcome during switching is larger in this case than in 180° domain switching, emphasising the pinning associated with bundle domain walls.
In summary, we demonstrated that true ferroelectric domain switching occurs when both 180° and bundle-domain switching take place simultaneously, but at different length-scales. This gives rise to a two-step switching process that comprises a fast KAI and a slow NLS behavior, while we interpreted the latter as an averaging of multiple KAI kinetic events from ordered depinning sites. In addition to the analysis discussed here, we note that our suggestion that a switching event occurs simultaneously at two time and length scales that span several orders of magnitude has been reported recently by Stolichnov et al. [18] Here, the authors reported that domain walls attract oxygen vacancies, which in turn give rise to electric conducting paths. When applying an external electric field for a short time (a few ns) the switched domain walls move. Yet, the trail or footprint of oxygen vacancies remains for a day or two before diffusing away. This activity of oxygen vacancies extends our analysis beyond ferroelectricity and ferroelasticity, to magnetic domain walls: In a perovskite titanate, the higher likelihood of the presence of oxygen vacancies in domain walls converts neighboring Ti +4 to Ti +3 , while Ti +3 is magnetically active. Therefore, there will be enhanced magnetism at the domain walls even in PZT. This magnetism interacts with the ferroelastic strain via magnetostriction of the form of E2H2 (or P2M2), producing a magnetodielectric response [19]. In these scenarios, many of the effects depend strongly on the oxygen vacancies and can therefore be reduced, e.g., by annealing in oxygen, allowing future examination of our analysis. Finally, combined with the recently reported controllable bundle domain reversal [26], the observed electromechanical coupling augmentation is expected to advance ferroic-based nano electro-mechanical systems, such as actuator, cellular antennae etc as well as nanoscale switching devices to lower-power and higher performance.