Exploring Vertex Interactions in Ferroelectric Flux-Closure Domains

Jul 24, 2014 - domain-wall vertex junctions prompt a consideration that vertex−vertex interactions could be influencing the measured kinetics. Howev...
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Exploring Vertex Interactions in Ferroelectric Flux-Closure Domains Raymond G. P. McQuaid,† Alexei Gruverman,‡ James F. Scott,§ and J. Marty Gregg*,† †

School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, U.K. Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of NebraskaLincoln, Lincoln, Nebraska 68588-0299, United States § Department of Physics, Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, U.K. ‡

ABSTRACT: Using piezoresponse force microscopy, we have observed the progressive development of ferroelectric fluxclosure domain structures and Landau−Kittel-type domain patterns, in 300 nm thick single-crystal BaTiO3 platelets. As the microstructural development proceeds, the rate of change of the domain configuration is seen to decrease exponentially. Nevertheless, domain wall velocities throughout are commensurate with creep processes in oxide ferroelectrics. Progressive screening of macroscopic destabilizing fields, primarily the surface-related depolarizing field, successfully describes the main features of the observed kinetics. Changes in the separation of domain-wall vertex junctions prompt a consideration that vertex−vertex interactions could be influencing the measured kinetics. However, the expected dynamic signatures associated with direct vertex−vertex interactions are not resolved. If present, our measurements confine the length scale for interaction between vertices to the order of a few hundred nanometers. KEYWORDS: BaTiO3, flux−closure, vertices, domain dynamics, ferroelectrics which they coalesce, leading to a distinct anomaly in the fielddriven dynamics. For ferroelectrics, whether or not proximate vertices are expected to have a mutual interaction at all is not as clear; if interactions do exist their nature and effects on dynamics are also unknown. In work carried out on BiFeO3 thin-films by Vasudevan et al.,10 vertex patterns are seen to be reversibly written/erased and have associated disclination defects that can be clearly seen in topographical maps; a stress-field interaction between neighboring disclination vertices could perhaps be envisaged, similar to the case in magnetics. In addition, a mounting body of static observations11−13 suggests that pairs of separated 3-fold vertices are more stable than more compact 4fold “core” vertex geometries, implying that any such vertex− vertex interaction in dynamic observations is likely to be repulsive in character (i.e., the vertices prefer to remain spatially separated). These ideas have a theoretical basis in early calculations by Srolovitz and Scott,14 based upon Potts and clock models, in which different vertex configurations could be favored depending on the dipole−dipole interaction energetics used. Specifically, they gave a phenomenological explanation of whether 4-fold vertices or adjacent pairs of 3-fold vertices are stable and the mechanism by which one would coalesce or separate into the other in ferroelectrics of a given symmetry. However, this work neither predicted which vertex arrangement would be stable in a particular real ferroelectric material, nor

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n ferroics research, emerging device concepts that exploit the functionalities associated with mobile structural interfaces has prompted a surge of interest in the mapping and understanding of domain walls.1−3 Although closely related, much less is known about the properties of domain-wall vertices (meeting points of two or more walls). For ferroelectrics research, although domain-wall vertices have long been considered as ubiquitous features of domain microstructure,4 it is only with the advent of dipole-resolution imaging techniques that an unexpected potential for topological complexity has been revealed. In particular, the most compelling evidence for elusive toroidal structures in ferroelectrics has to date been observed at nanoscale vertices, via local dipole mapping using aberration-corrected transmission electron microscopy (TEM).5,6 Ferroelectric vertices can also exhibit unusually high electrical conductivity normal to the plane in which they lie,7 revealing that they can display functional characteristics distinct from the surrounding domain walls. To date, however, virtually nothing is known of how these vertices are expected to behave in dynamic observations. Experimentally, the static and dynamic interactions of vertices in magnetic systems are somewhat better established and they may help inform our expectations for ferroelectric analogues. In magnetism, proximate vertices are known to interact via disclination stress-fields with the result that certain vertex-pairs are seen to be preferentially stabilized.8 In fluxclosing domain patterns, where vertices develop naturally due to the domain topology, Masseboeuf et al.9 have revealed the influence of vertex−vertex interactions on dynamical processes. They identify a critical-separation between 3-fold vertices at © 2014 American Chemical Society

Received: February 21, 2014 Revised: June 13, 2014 Published: July 24, 2014 4230

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Figure 1. (a) Lateral-mode piezoresponse amplitude images revealing growth development of quadrant flux-closure domains after removal of an inplane electric field-pulse, E (direction indicated by arrow). Sense of time elapsing increases from image i to iv. (b) Vertex separation, x(t), plotted as a function of time (after delivery of poling set-pulse at t = 0). Data points corresponding to image panels i−iv in part a are labeled. A schematic labeling the vertex separation x(t) and the in-plane polar domain structure is shown as an inset. (c) loge x(t) as a function of time using the data plotted in part b along with linear fit (dashed line). Inset shows relative vertex velocity dx(t)/dt as a function of vertex separation x(t) with linear fit (dashed line).

gave any estimate of the coalescence times, nor any details of the collision dynamics. To date, relevant experiments to allow further development in insight have not been forthcoming. In this letter, we investigate the role of ferroelectric vertex− vertex interactions in dynamic processes by using piezoresponse force microscopy (PFM) to monitor the time-varying separation of a pair of 3-fold domain wall vertices. The vertex pair appears naturally as part of flux-closure domain structures that develop in thin single-crystal platelets of BaTiO3, under the influence of residual depolarizing fields that develop after electric-field poling. Ultimately, the purpose of this investigation is to determine the influence of vertex−vertex interactions on the relaxation kinetics of the system as it tends toward the flux-closure state. Single- and double-closure patterns are observed and we identify growth kinetics with a rate that decreases exponentially in time. The process occurs slowly, on a time scale of hours, with an average velocity of order ∼1 nm/s that is commensurate with creep processes in oxide ferroelectrics.15 Initial inspection suggests that these kinetics can be rationalized by a picture involving progressive screening of macroscopic destabilizing fields, primarily surfacerelated depolarizing fields. However, in some cases we also observe a highly correlated redistribution of the vertex pattern, seen to proceed by coalescence of a connected pair of 3-fold vertices followed by their reseparation at 90° to the original orientation. Here, the specific domain-wall vertex patterns observed and their evolution is exactly as discussed in Srolovitz and Scott’s vertex models, prompting us to examine the possibility that direct vertex−vertex interaction phenomena could also be an important driving factor in determining the

growth kinetics. On this basis, we consider a scenario where direct vertex−vertex interactions instead determine the observed kinetics, with the interaction modeled as having the character of an overdamped oscillator system. However, this picture is unable to consistently explain the range of different flux-closure vertex patterns observed and we do not resolve any anomalies in our kinetics data associated with vertex−vertex interactions (as expected from parallel studies in magnetism). The resolution limitations of our scanning probe based measurements do, however, allow us to estimate the spatial range of any short-range interaction between vertices (if it occurs) to within a few hundred nanometers. Results. Single- and double-flux-closure domain patterns are consistently seen to form in a single-crystal BaTiO3 platelet measuring 6 × 4 × 0.3 μm after removal of a poling voltage setpulse. Fabrication and experimental approach details can be found in the Experimental Methodology. The precise closure configuration that develops in this platelet can vary and is somewhat unpredictable. In a previous report we identified that these closure patterns only develop along 180°-type boundaries, hence which pattern ultimately forms is likely related to the initial number and position of backswitched 180°-type domains that appear during the early stages of relaxation. The typical case where a single closure-loop develops under the influence of residual depolarizing fields is presented in Figure 1a. The time evolution of the remnant domain pattern is imaged by PFM after set-pulse poling (10 Vdc applied across planar electrodes for 2 min). Four mesoscale packets of fineperiod a/c ferroelastic domains can be identified (the ferroelastic domain modulation in the left/right quadrants is 4231

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Figure 2. Lateral-mode piezoresponse amplitude (a) and phase (b) maps of a single closure pattern in a rectangular platelet after removal of a prepoling field pulse (in direction of arrow labeled E). Lateral-mode piezoresponse amplitude (d) and phase (e) of the same specimen after 231 h have elapsed. (c),(e) Schematic illustrations of the vertex patterns in parts a and d. Arrows denote in-plane polarization orientation P within each quadrant. Edep indicates direction of depolarizing field inferred from labeled schematic uncompensated surface charges.

Figure 3. (a) Time sequence of lateral piezoresponse maps showing explicitly how the vertices approach, coalesce and separate after delivery of a prepoling field pulse (in direction labeled by arrow, E). Schematic illustrations of the instantaneous boundary configuration during the process are also shown, right. Red circles indicate location of domain wall vertices. Arrows within each quadrant denote polarization direction P. (b) Plot of vertex separation x(t) as a function of time. Blue region corresponds to orientation of blue colored boundary in schematic (a), top, green region to that of green colored boundary in schematic a, bottom. (c) loge x(t) as a function of time, and linear fit, using the data plotted in part b.

subtle for this scan orientation in LPFM mode) and the mesoscale quadrant boundary pattern is seen to be connected by two 3-fold vertices at either end of the 180°-type boundary (illustrated schematically in Figure 1b inset). As time elapses,

on the scale of hours, the top and bottom quadrants are seen to increase in area and the separation between the two 3-fold vertices is seen to decrease. We choose to monitor the development of the closure structure by plotting the vertex 4232

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Figure 4. (a) Collage of piezoresponse images (obtained with scanning probe cantilever aligned along ⟨110⟩pc direction) showing the complete ferroelastic stripe domain topology of a double closure geometry. The direction of the prepoling field pulse, E, is indicated. (b) In-plane vector-PFM phase map: arrows denote in-plane polarization direction, P. (c) Lateral piezoresponse maps of a separate observation of double-closure pattern ∼20 min after initial poling (top) and ∼7 h later (bottom). (d) Vertex separation x(t) versus time for the left-hand quadrant pattern in part c after removal of poling set-pulse at t = 0. The plotted length x(t) is labeled in part c and the data points extracted from part c are labeled (i and ii). Inset schematic shows the polar domain structure of the double-closure pattern.

separation as a function of time, x(t), presented in Figure 1b. Each point in the plot corresponds to a single PFM image, each of which takes approximately 10 min to record. We find that the development of these flux-closure domains (monitored by vertex separation x(t)) displays exponential rate kinetics over the measured trajectory, evidenced by the linear form of logex(t) versus time (Figure 1c). Vertex separation x(t) is seen to follow the form x(t ) = xf + (x0 − xf )e−t / c

experiment is repeated (i.e., the platelet is poled and then allowed to relax) while continuous PFM scanning is carried out. By this means, we have been able to observe directly in realtime that the 90° rotation of the domain pattern occurs by a two-step mechanism of boundary reconfiguration, as shown in Figure 3a. At early times, the top and bottom quadrants are seen to grow in size accompanied by a decrease in the separation of the 3-fold vertices (just as in Figure 1). After ∼3 h have elapsed, the vertices come together and the adjoining 180°-type boundary is annihilated, suggesting (within PFM resolution limits) that a single 4-fold vertex remains (Figure 3a, middle panels). The contact area between the top and bottom quadrants is seen to increase further, accompanied by reappearance of a pair of 3-fold vertices positioned at 90° to their original configuration (Figure 3a, bottom panels). The kinetics of this process are plotted in Figure 3b with no further change from the final pattern being recorded over a further observation period of 17 h. Just as for the observations in Figure 1, we also find that the vertex separation displays exponential kinetics over most of the measured trajectory, evidenced by the linear relationship between loge x(t) as a function of time (t) plotted in Figure 3c. We note that the 3 h time scale for vertex coalescence in Figure 3 is significantly shorter than in the Figure 2 observation. This could perhaps be due to additional voltage training of the sample (through the planar electrodes) prior to the observations made in Figure 3 or the influence of continuous rastoring of the biased scanning probe over the platelet in this case. The final scenario that we observe (following the same poling procedure) is the development of double-closure patterns (Figure 4a,b). If two 180°-type domain walls are stabilized in the material during the early stages of relaxation

(1)

where x0 is the initial vertex separation at t = 0, xf is the final vertex separation at t = ∞, and c is a characteristic time for the process. This behavior also inherently suggests linear behavior in the time derivative dx(t)/dt with respect to the vertex separation x(t), where dx(t ) ∝ −(x(t ) − xf ) dt

(2)

and is plotted as an inset in Figure 1c. A more unusual, lesscommon observation involving the development of a single closure pattern is shown in Figure 2. In this separate observation, a PFM domain map captured 26 h after initial poling shows a single closure pattern at the left-hand side of the platelet, similar to observations made in Figure 1. The domain boundary configuration in the same specimen is shown after a further 231 h have elapsed (with no intermittent PFM scanning carried out in between) revealing that the domain geometry has undergone significant transformation. It appears that the vertex pattern shown in Figure 2d−f, although quite asymmetric, has rotated ∼90° relative to the original pattern (Figure 2a−c) at some point after the initial observation. In order to try and reveal the process by which this transformation occurs, the 4233

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after poling, flux-closing domains can develop along both of these boundaries, as shown in Figure 4c. The final result is selfordered closure circuits that have polarization arrangements that rotate in opposite senses, as revealed by in-plane vectorPFM (Figure 4b). The growth kinetics (Figure 4d) show a similar profile to that observed for a single isolated closure pattern (Figures 1 and 3) although no vertex-coalescence is observed in this scenario. At the mesoscale boundary level this structure is the archetypical flux-closed Landau−Lifschitz stripe domain pattern16 that has long been a commonplace observation in magnets across several length scales17,18 but predicted exclusively at the nanoscale in ferroelectrics.19,20 As with the single-closure observations, the fine-scale ferroelastic domain substructure helps to offset the elastic energy penalty normally associated with quadrant domain patterns, allowing them to be stabilized on the mesoscale. We now consider in turn two different pictures that can capture certain features of the observed relaxation kinetics. Since flux-closure patterns are expected to form in response to depolarizing fields, we first consider growth kinetics determined by progressive screening of a thermodynamic driving field. Second, we consider an alternative scenario where the kinetics are instead determined primarily by direct vertex−vertex interaction effects. Crucial to both these pictures is that they can reproduce the observed exponential growth kinetics. Macroscopic Depolarizing-/Stress-Field Driven Kinetics. In our experiments, the platelet starts from a fully poled state (as illustrated in Figure 3 of ref 21) and ends in a flux-closed domain pattern, either consisting of a single-closure (Figure 1) or double-closure arrangement (Figure 4). This is a strong indicator that at the moment the poling-field is removed there is a residual depolarizing field associated with the instantaneous switched domain pattern. In ferroelectrics this most commonly drives the formation of 180° stripe domains but in this special case instead drives the formation of 90° flux-closure domains (possible because of the strain-compensating ferroelastic domains within each quadrant). Crucially, we anticipate that the rate of development of flux-closure domains will be proportional to how far the system is from the final equilibrium pattern. With reference to Figure 5a we label the flux-closed length measured at a given time “y(t)” (i.e., the length of the base of the closure-triangle along the top long-edge of the platelet), the final measured flux-closed length when no further development is observed “yf ” (where y(∞) = yf), and the width of the platelet “w”. Focused ion beam (FIB) design specification of the platelet geometry sets the value for w and places an upper-limit on yf as the long-edge platelet length, L. The rate of increase of flux-closed length can be expressed as follows: rate of flux-closure development ∝ remaining amount of uncompensated surface charge, i.e. dy(t ) 1 = (yf − y(t )) dt τ −1

where τ

Figure 5. (a) Schematic illustration of the quadrant boundary configuration corresponding to x(t) < 0 in the depolarizing field model. The length of the real observed vertex separation in the horizontal plane (red line) is seen to be equivalent to the separation between the projected apexes of the quadrant triangles in the vertical plane (black double-headed dashed arrow). The length and width of the platelet are labeled as L and w respectively. The instantaneous fluxclosed length at time t is labeled as y(t). (b) x(t) data from Figure 3b is replotted with the best fit linear trend generated using the depolarizing field model.

specimen edges), a simple geometrical relation between the amount of flux closed length, y(t), and the expected vertex separation, x(t), can be established: x(t ) = w − yf (1 − e−t/ τ )

Additionally, if at t = 0 the vertices are initially generated at the opposite edges of the platelet (as the data suggests) then the initial vertex separation is simply equal to the width of the platelet; i.e., x0 = w. Equation 5 shows that, in principle, the exponential kinetics inferred from the time dependence of vertex separation (eq 1) can have its origin in domain growth associated with progressive screening of a depolarizing field. A curious feature of eq 5 is that if the platelet geometry is such that L > w (as in our experiments) then it is possible for yf > w such that the x(t) function can predict negative values for the vertex separation. This seemingly nonintuitive result just describes the continued growth of the top/bottom closure triangles after they impinge upon each other (for scenarios when vertex coalescence is observed). The geometric interpretation of negative x(t) values is schematically illustrated in Figure 5a showing that they describe the separation between the projected vertices (dotted lines in Figure 5a) of the intersecting top/bottom quadrant triangles along the same axis as before coalescence. Under the assumption that the quadrant boundaries remain oriented at 45° to the platelet edges, this length is geometrically equivalent to the real vertex length measured at 90° to this axis. The linear fit plotted in Figure 5b (based on the data presented in Figure 3b) confirms that eq 5 can represent the data from positive through to negative x(t) values in a single continuous function even though the implicit assumption of ⟨110⟩pc aligned quadrant boundaries is not

(3)

is a constant of proportionality. Hence

y(t ) = yf (1 − e−t/ τ )

(5)

(4)

if we assume that at t = 0 (the moment the poling voltage is removed) the amount of flux closed length along the top/ bottom edges is zero (i.e., y(0) = 0). If we make the simplifying assumptions that the top and bottom triangular flux-closure domains grow at the same rate and maintain quadrant boundaries aligned along ⟨110⟩pseudocubic (pc) (i.e., at 45° to the 4234

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strictly true. This means that the simple depolarizing field model can still be used to account for the domain vertex reconfiguration phenomena observed in Figures 2 and 3. If the measured platelet width (w = 4 μm) is entered into eq 5 as a fixed parameter, the data fits in Figure 5b return a characteristic time of 1.8 h and an estimated flux-closed length of yf = 4.5 μm. In Figure 3 the top/bottom edges of the platelet are not within the scanned area, however the visible flux-closed length along the top/bottom of panel a(iii) measures ∼2.9 μm and represents a lower bound to yf that is in reasonable accord with the model prediction. In Figure 1a, panel iv, where the entire platelet is scanned, the measured yf = 4.5 ± 2 μm is in good agreement with the model and is close to the upper-limit set by the platelet length of L ≈ 6 μm. On average, the measured value for lf associated with a single closure pattern is ∼3 μm (since often two closure-patterns form in the same 6 μm long platelet), however this value has been seen to range from a minimum of ∼1 μm to a maximum of ∼4.5 μm. Direct Vertex−Vertex Interaction Driven Kinetics. Recognizing that the mesoscale vertex patterns presented in Figures 2 and 3 are qualitatively the same as those of Srolovitz and Scott’s14 model predictions suggests that, in addition to depolarizing fields, stabilization of specific vertex patterns could be an important factor in determining growth kinetics. Following their predictions, observation of pairwise coalescence of 3-fold vertices, such as is seen in the early stages of our experiments (Figure 2a-c, Figure 3a), should end with formation of a single stable 4-fold vertex. However, our Figure 3 data reveals that this 4-fold vertex structure is in fact only metastable and subsequently breaks-up again into a 3-fold vertex pair that is rotated ∼90° relative to the original distribution. The observation that the system seems reluctant to adopt a 4fold vertex state is in line with a growing body of evidence in ferroelectrics research that 4-fold “core” junctions are in general avoided in favor of 3-fold wall junctions,5,6,12 and even in hierarchical domain structures (see ref 11 and Figure 2 of ref 13). While Srolovitz and Scott’s models can go some length toward rationalizing the observed boundary configurations, they do not offer any predictions for the associated kinetics. In this regard, we may take some direction from selected studies in magnetism where analogous vertex approach, coalescence and scattering phenomena are observed. In particular, the dynamic evolution of the vertex patterns observed here mirrors exactly the process of magnetic soliton coalescence and scattering modeled by Komineas and co-workers22,23 (and more generally by Zakrzewski et al.24) if we tentatively draw analogy between the ferroelectric vertices and the magnetic solitons. In this description, two spatially separated magnetic solitons approach, collide head-on to form a bound metastable state, and are then ejected again at 90° to the incident trajectory.22 The same particle-like scattering analogy of head-on collision, coalescence, and 90° ejection, can be applied explicitly to the ferroelectric domain wall vertices in Figure 3a. Because the nanoscale dipolar character of ferroelectric vertices has been seen in some cases to exhibit real continuous polar rotation,5,6 as is a prerequisite for the hypothesized ferroelectric dipole vortex, this comparison to magnetic vortex structures may be more than just superficial. To simplify the analysis for our data, we can consider the extreme scenario where the kinetics are determined entirely by a vertex−vertex interaction and depolarizing fields are negligible. In this picture, we assume that when the poling field is removed the vertices are created

beyond their equilibrium separation and start to relax, leading to their mutual approach and eventual coalescence. On this basis, we find that the characteristic exponential kinetics profile can be reproduced with the ansatz that the vertex−vertex interaction exhibits a spring-like behavior. Assumption of an effective medium viscosity is required to account for the decreasing rate of vertex approach, such as by the phonon-drag mechanism described in ref 25. The standard equation of motion for a damped harmonic oscillator is given as26 x(̈ t ) + 2βx(̇ t ) + ω0 2x(t ) = 0

(6)

where β = b/2m, b is a damping parameter, m is traditionally associated with the mass of the spring, and the natural frequency ω0 = (k/m)1/2, where k is the spring constant. We picture that, as soon as the initial poling field is removed, the vertices are created with a spatial separation x0 and start to move from rest toward each other in a manner similar to a stretched spring relaxing in a viscous medium. Under these initial conditions we can derive an expression for the vertex separation as a function of time, x(t), based on standard solutions for an overdamped oscillator, having the following form: x0 − xf −At x0 − xf −Bt e + e x(t ) = xf + (7) 1 − A /B 1 − B/A Here A = β − (β2 − ω02)1/2, B = β + (β2 − ω02)1/2. Under conditions of heavy damping where β ≫ ω0, and therefore B ≫ A, eq 7 approximates as x(t ) ≈ xf + (x0 − xf )e−At

(8)

Equation 8 has the same exponential form as the expression in eq 1 suggesting that it can characterize the measured kinetics data presented in Figures 1b and 3b. Discussion. In order to determine which of these models is more appropriate, we can investigate how well each rationalizes the range of experimentally observed scenarios. In the depolarizing field model, the redistribution of vertices can be seen to occur as a byproduct of flux-closure domain growth. Therefore, when flux-closure is achieved the vertex motion is expected to cease. In this way the vertex coalescence and reseparation phenomena seen in Figure 2 can be rationalized due to the single-closure pattern growing in size in an attempt to maximize the amount of flux-closed surface (the reduction in uncompensated charge is schematized in Figure 2, parts c and f). In contrast, the fact that no vertex coalescence and reseparation is observed for the double closure pattern in Figure 4 can also be appreciated by the same logic. Here, the growth of the two closure patterns would be expected to cease when the entire surface perimeter is flux-closed; experimentally, this is seen to be achieved before the vertices coalescence and they remain separated thereafter as a result. However, the line of reasoning that the growth kinetics and stabilized vertex patterns are entirely driven by the need to obviate a depolarizing field does not appear to be a complete description either when we look closer at Figure 1. The earliest captured domain map (Figure 1a(i)) shows that domains with in-plane polarization component parallel to the surface have quickly formed in order to minimize the amount of stray-flux associated with the top/ bottom sample edges. This indicates that further changes in the domain wall configuration are likely not driven by depolarizing fields during the subsequent time window of PFM observation (Figure 1a(ii−iv)). What is clear, is that the top and bottom 4235

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kinetics of the system. This aside, purely symmetry based arguments suggest that the initial and final 90° rotated 3-fold vertex patterns are symmetrically equivalent in the tetragonal BaTiO3 system studied, and would therefore be thermodynamically degenerate. This makes it clear that other macroscopic thermodynamic driving fields (depolarizing-/stress-fields) are essential to the description and that direct vertex interactions and reconfiguration are secondary effects. This is mirrored in Masseboeuf et al.’s9 magnetics study where subtle flux-closure vertex−vertex interactions are seen to be superimposed on macroscopic field-driven dynamics. Conclusion. In conclusion, we have investigated the kinetics of apparent domain-wall vertex approach, coalescence, and 90° scattering that occur during evolution of a flux closure domain structures in a single crystalline platelet of BaTiO3. The vertex patterns observed are precisely those predicted in statistical mechanical models by Srolovitz and Scott and the specific growth modes have been seen to bear strong similarities to those of soliton collision models developed by the magnetics community. However, a picture where microstructural change was driven solely by vertex−vertex interaction seemed unable to capture all the features of the observed kinetics in a consistent way. Progressive screening of a macroscopic depolarizing-field, and possibly stress-fields, was seen to well characterize the observed broad-brush kinetics where vertex− vertex interactions could play a secondary role. Even then, no signature of direct vertex−vertex interaction was observed in the measured kinetic profile. This suggests that vertex−vertex interactions in ferroelectrics are subtle and that more detailed experiments with improved spatial and time resolution are essential in order to determine unambiguously their role in dynamical processes. Experimental Methodology. A thin {100}pc-faced slice of BaTiO3 (measuring approximately 6 × 4 × 0.3 μm thick) with edges parallel to ⟨100⟩pc was machined from a commercially obtained bulk single crystal using FIB techniques and then incorporated into a prepatterned planar electroded device according to a previously established methodology.31,32 The electrode design is specifically tailored to allow passive PFM monitoring of the instantaneous domain configurations that develop after delivery of planar electric-field pulses. Prior to the observations made in Figures 1−4, a 10 Vdc bias was applied (ex situ) across the 3 μm interelectrode gap, parallel to ⟨100⟩pc, for 2 min. After removing the bias, the platelet was mounted in the PFM system so that subsequent relaxation of the remnant domain structure could be passively mapped as a function of time. In Figure 2, PFM mapping 26 h after removal of bias and the subsequent state after a further 231 h had elapsed were captured with no intermittent scanning. In Figures 1, 3, and 4, PFM rastoring is carried out at multiple intervals between the initial and final captured states. In all graphs of vertex position, x(t), versus time, t, (Figures 1,3, and 4) each plotted data point is extracted from a single captured PFM image. The minimum separation between data points corresponds to the PFM scan acquisition time of 10−20 min depending on scan rate. All experiments in this study were carried out on the same platelet although formation of single flux-closure structures has been previously observed in other samples made to a similar specification, as described in ref 21.

domain quadrants are increasing in area and there is a change in the inclination of the mesoscale quadrant boundaries closer to approximately ⟨110⟩pc (45° to platelet edges). These further changes can be rationalized using the domain wall energetics arguments that form the basis for Kittel domain scaling,27 i.e., with considerations toward achieving boundaries that are electrostatically uncharged (i.e., along {110}pc planes), are mechanically compatible, and have minimized length. So, while the initial rapid growth of the closure structures almost certainly serves to obviate depolarizing fields, in reality the observed long-term kinetics of the domain pattern also involves these additional processes. By taking into account these energy considerations, in addition to the depolarisation energetics of eq 3, a more complex expression that more completely accommodates these kinetics data could be derived, using the general approach outlined in ref 27. In principle, these mechanisms could generate our measured kinetics profile on the basis that the rate of change of the microstructure toward the thermodynamically stable state will be proportional to the amount of material for which microstructural rearrangement has not yet occurred; these conditions naturally generate an exponential profile in time. The observations in Figure 1 would suggest that the characteristic time for domain reconfiguration associated with these secondary processes is longer than for that associated with screening depolarizing fields. In this regard, we note the comparably slow creep-like growth rates seen for elastically driven nanodomain faceting phenomena in PbZrTiO328 and polyvinylidene fluoride29 thin films (minutes to hours). These data suggest that residual strains associated with ferroelastic domain rearrangement in our experiments may also be involved in determining the long-term kinetics. For cases where depolarizing fields seem unable to fully account for the observed vertex reconfiguration we note that the modeled stress-field patterns seen to develop in sheared rectangular plates30 could also be related. Nonetheless, while there may be a drive to stabilize a specific vertex pattern, as detailed by Srolovitz and Scott, there are several reasons why it is very unlikely to be the primary factor (instead of depolarizing fields) causing microstructural change in the system. First, in the modeled scenarios studied by Komineas and co-workers22,23 and in magnetics experiments by Masseboeuf et al.,9 anomalies in the dynamics are observed in the vicinity of the coalescence event, as a signature of a short-range mutual attraction. In Figure 3b of our study no such anomaly in this region of the data is obvious: within uncertainty, a smooth monotonic exponential profile is measured. However, it is a possibility that such an effect could be subtle and require measurements with superior resolution (especially considering the nanometer length scales and millisecond time scales involved in Masseboeuf et al.’s observations). Our data allow us to identify a vertex separation on the order of a few hundred nanometers over which any such mutual short-range interaction may be active, if present (based in Figure 3b data). Another issue arises due to the fact that there is a degree of stochasticity in the way in which the vertex pattern is ultimately stabilized. Sometimes the entire sequence of vertex-pair approach, coalescence and reseparation is observed (Figures 2 and 3) whereas in other cases, e.g., in double-closure pattern observations, the initial 3-fold vertex patterns become “locked-in” and no vertex coalescence is observed (Figure 4). This fact is difficult to reconcile in a picture where direct interaction and reconfiguration of vertices is considered the primary underlying thermodynamic factor driving the relaxation

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AUTHOR INFORMATION

Corresponding Author

*(J.M.G.) E-mail: [email protected]. 4236

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Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

The authors acknowledge financial support from the Engineering and Physical Sciences Research Council (EPSRC) and the National Science Foundation (NSF) under the Materials World Network (MWN) program (NSF Grant No. DMR-1007943 and EPSRC Grant No. EP/H047093/1). We thank the Leverhulme Trust for international network funding (F/00 203/V). R.G.P.M., and J.M.G. acknowledge support from the Department of Employment and Learning (DEL). R.G.P.M. acknowledges support from the Dunville Studentship scheme. A.G. acknowledges support from the NSF Materials Research Science and Engineering Center (Grant No. MRSEC DMR0820521). The authors acknowledge useful discussions with G. Catalan, P. Zubko, S. Felton, S. Komineas and participants of the meeting “Domain microstructure and dynamics in magnetic elements”, Heraklion, Crete, April 2013.

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dx.doi.org/10.1021/nl5006788 | Nano Lett. 2014, 14, 4230−4237