Hidden domain boundary dynamics toward crystalline perfection

Significance

A central goal of materials science is to understand the pathways by which materials transform. The dynamics of these processes involve heterogeneity, disorder, and irreversibility, and underlie a range of nonequilibrium phenomena with numerous technological applications. A key obstacle toward understanding these transitions is visualization of the associated nano and mesoscale dynamics, which are often distinct from those at the atomic scale and occur over multiple timescales. Here, we reveal the nonequilibrium, irreversible, and statistical dynamics of defects during a light-induced phase transition, combining single-shot pump–probe X-ray photon correlation spectroscopy and coarse-grained modeling. Our work enables visualization of the nanoscale nucleation processes and the motion of domain boundaries that are dynamically created and then disappear, revealing pathways toward crystalline perfection.

Abstract

A central paradigm of nonequilibrium physics concerns the dynamics of heterogeneity and disorder, impacting processes ranging from the behavior of glasses to the emergent functionality of active matter. Understanding these complex mesoscopic systems requires probing the microscopic trajectories associated with irreversible processes, the role of fluctuations and entropy growth, and the timescales on which nonequilibrium responses are ultimately maintained. Approaches that illuminate these processes in model systems may enable a more general understanding of other heterogeneous nonequilibrium phenomena, and potentially define ultimate speed and energy cost limits for information processing technologies. Here, we apply ultrafast single-shot X-ray photon correlation spectroscopy to resolve the nonequilibrium, heterogeneous, and irreversible mesoscale dynamics during a light-induced phase transition in a (PbTiO₃)₁₆/(SrTiO₃)₁₆ superlattice. Such ferroelectric superlattice systems are a useful platform to study phase transitions and topological dynamics due to their high degree of tunability. This provides an approach for capturing the nucleation of the light-induced phase, the formation of transient mesoscale defects at the boundaries of the nuclei, and the eventual annihilation of these defects, even in systems with complex polarization topologies. We identify a nonequilibrium correlation response spanning >10 orders of magnitude in timescales, with multistep behavior similar to the plateaus observed in supercooled liquids and glasses. We further show how the observed time-dependent long-time correlations can be understood in terms of stochastic and non-Markovian dynamics of domain walls, encoded in waiting-time distributions with power-law tails. This work defines possibilities for probing the nonequilibrium and correlated dynamics of disordered and heterogeneous media.

While there have been numerous advances in approaches to drive matter into novel metastable phases and across phase boundaries, few of these approaches provide for direct visualization of the intrinsic dynamical heterogeneity and nonequilibrium response that often underlies these processes (1–4). Nonequilibrium states are not, as often thought, always short-lived or negligible after some brief waiting time (5). As a prototypical example, consider the process of rapidly tuning matter across a phase boundary to a symmetry-broken phase. This corresponds to an ultrafast quench in which phases nucleate and grow in a coarsening, self-similar response described by a power law, essentially enforcing nonequilibrium response and ergodicity breaking to long times (5). Understanding these universal processes requires experimental tools and approaches which probe the intrinsic role of fluctuations, disorder, heterogeneity, and memory of initial conditions (6–9), and not just the average structure as probed by crystallographic techniques (10). Similarly, most classical nucleation and growth models focus on the volume fraction of the transformed phase, neglecting heterogeneities associated with interfaces or domain walls (11). Phenomenological models have been developed for such phenomena arising from time-dependent Landau–Ginzburg functionals, which can describe these nonlinear, self-similar dynamical responses (12, 13). But theoretical models which capture the nonequilibrium response of the defects themselves (e.g., the nonequilibrium dynamics within a domain wall and their evolution under external stimuli) remain to be fully developed. From a broad perspective, defect–defect interactions (e.g., domain–wall–domain–wall interactions or other types of correlated motion) are poorly understood but underlie a vast range of physics and technological applications (14, 15).

Although state-of-the-art time-resolved X-ray or electron scattering may be used to probe heterogeneity in samples with phase boundaries (16), such approaches largely neglect these defect states and assume a coherent response across all unit cells, or perform an effective ensemble average, thus blurring variations in the local structure and the presence of phase boundaries (1–3, 10, 17). Defects do disrupt the long-range atomic and structural order that produces Bragg spots and satellite peaks and therefore change the diffraction lineshape. However, these modulations are subtle and often difficult to resolve (18–20). Instead, approaches capable of probing the small volume fraction associated with the boundaries of domains (i.e., domain walls and other defect states) and more broadly, the microscopic trajectories that a heterogeneous material follows through configuration space as it transforms are required. X-ray photon correlation spectroscopy (XPCS) offers a method of monitoring the dynamic heterogeneity inherent in phase changes, but is most often applied to probe dynamics on much slower time-scales than investigated here (21, 22). Under coherent X-ray illumination, the temporal evolution of nano/mesoscale features gives rise to time-varying modulations to the diffraction pattern (23, 24), enabling approaches for correlating diffraction images. When tracking the correlation of the diffracted spot rather than the integrated intensity, our experiments indicate a complex set of dynamics extending to time-scales of order 100 ms, whereas the dynamics associated with the integrated intensity are largely complete after a few microseconds. In this dramatic difference lies hidden information about interfaces and defects that are created, their subsequent interactions, and their eventual annihilation during the transformation as the system moves toward crystalline perfection.

(PbTiO₃)_n/(SrTiO₃)_n superlattices offer a platform to experiment with previously unexplored mesoscale-phase transitions because they exhibit a wide array of topological-polar orders, ranging from skyrmions to vortices and merons, with interesting mesoscale features (25–28). Previous work has shown the existence of a long-lived metastable vortex supercrystal (VSC) phase only accessible through excitation by ultrafast pulses (29). Fig. 1A shows reciprocal space mapping in the Q_z–Q_y plane about the pseudocubic 002-diffraction peak of the superlattices and the DyScO₃ substrates upon which they are grown before and after irradiation by a single 100 femtosecond (fs) pulse of 400 nm light, showing evidence for a solid–solid structural phase transition. Previous studies on this phase transition have shown that under optical excitation, the pristine polarization structure disappears on ultrafast timescales. The VSC emerges on nanosecond time scales until the satellite peak intensity saturates around 1 µs. This ferroelectric order changes due to complex interplay between the ionic displacement, free charges induced by the optical pump and rise in temperature. While these studies did analyze the pathway through the transition, they did not describe the emergence of a mesoscale domain structure (30).

Fig. 1.

(A) Reciprocal space maps of pristine and transformed structures. The red box shows area of reciprocal space shown in B. (B) Single-shot XPCS setup showing the pulse-train sequence (Top) and sample diffraction images (Bottom) capturing pristine, transient intermediate (at ΔT = 1 μs), and final state speckle patterns corresponding to boxed area in A. Black boxes show the region of the satellite peak shown in C. (C) Representative single location 50 × 50-pixel regions (centered at the peak centers extracted by procedure detailed in Materials and Methods) at the center of the transient and final supercrystal satellite peak at selected ΔT values. Whereas the integrated intensity of the entire peak saturates after tens of microseconds, the correlations in the speckle patterns continue to evolve on millisecond timescales.

We conducted a single-shot optical pump-X-ray probe XPCS experiment at the Linac Coherent Light Source at the SLAC National Accelerator Laboratory to monitor the nonequilibrium speckle evolution during this irreversible light-induced phase transition (Materials and Methods). A sequence of 30 Hz repetition-rate, 9.8 keV probe X-ray pulses with ~50 fs pulse widths was incident on the sample (labeled in Fig. 1B by index n) with a single 400 nm pump pulse (50 fs duration) arriving at controllable time delays relative to one of the central X-ray pulses in the train. Thus, we obtained coherent speckle pattern snapshots of the equilibrium state before exposure (n < 0), the transiently evolving intermediate structure (n = 0), and the final state after the transition is complete (n > 0), which can be correlated relative to each other. Because the transformation is irreversible, after each sequence of pulses associated with a single pump pulse, the sample was translated to a new spot and the sequence was repeated. One such sequence obtained at a single spot on the sample is shown in SI Appendix, Fig. S1. The variation in intensity of the n > 0 VSC peak in SI Appendix, Fig. S1 shows the intrinsic fluctuations in the incident probe intensity inherent to the source. A sequence of transient (n = 0) shots from different spots on the sample are shown in SI Appendix, Fig. S2. The resulting speckle correlation was obtained by averaging the correlation functions from each sequence. The orientation of the sample was such that the first-order satellite peak associated with the VSC phase emerges as the transformation progresses (Fig. 1B).

Results

Fig. 1C shows the evolution of the single-shot speckle pattern as a function of time between the optical excitation pulse and the X-ray probe pulse (corresponding to n = 0, Fig. 1B); with the bottom row showing the final state after the transformation is complete (e.g., at ${t=t}_{\infty }$). Each time corresponds to a different location on the sample and the experiment samples approximately 20 spots at each delay. Qualitatively, whereas the integrated intensity reaches 90% of its final value on timescales of order ~10 µs, correlations in the speckle pattern comparing the transient shot to the final state continue to evolve on hundreds of millisecond timescales.

These speckle patterns encode information about the mesoscale structure of the induced supercrystal phase as it nucleates, complementary to the information that could be understood from the shape and position of the diffraction peak. We quantify the correlation of images at two different times using a pixel-by-pixel average to compute the time dependence of the correlation, using a two-time correlation function given by (31)

$$C\left(t_1,t_2\right)=\frac{\overline{I\left(t_1\right)I\left(t_2\right)}}{\overline{I\left(t_1\right)}\overline{I\left(t_2\right)}},$$ [1]

where I is the scattering intensity captured on a two-dimensional detector in a region of reciprocal space, t₁ and t₂ denote the times X-ray probe pulses arrive at the sample relative to the optical pump pulse, and the bars indicate an average taken over a region of interest (ROI) in momentum (Q) space. This can be thought of as a form of time-resolved correlation spectroscopy probing correlations between speckle patterns recorded at different time delays relative to the optical-excitation pulse. Additional correlation analysis was performed between postcharacterizations pulses (n > 0) to understand the very long-time (>3 ms) behavior of the sample. The final value of the correlation (i.e., at ${t=t}_{\infty }$) was calculated by averaging the correlation value between images n > 7. This correlation function has minimum value one for uncorrelated scattering patterns (31). A normalized quantity

$$\frac{\mathrm{\Delta }C}{〈C\left(f,f\right)〉}=\frac{〈C\left(f,f\right)-C\left(t,f\right)〉}{〈C\left(f,f\right)〉},$$ [2]

was also calculated, with t denoting the transient speckle pattern collected at a specific time delay, f denoting the final speckle pattern corresponding to ${t=t}_{\infty }$, and the angle brackets denoting an average across sample spots. This quantity corrects for the spot-to-spot variation in speckle contrast due to slight sample variations and X-ray beam quality differences (Materials and Methods).

In the following, we focus on the correlation between the transient intermediate structure at time t and the final-state structure at $t=t_{\infty }$, as encoded in $C\left(t,f\right)$. This tracks the pathway of the nonequilibrium transient phase toward the final equilibrium state. Fig. 2A shows the full two-time correlation function at two relative transient time delays averaged across sample locations, correlating each X-ray shot in the train with every other shot. The upper right quadrant correlates final shots with other final shots and is therefore significantly brighter, also showing that the final state is evolving over many orders of magnitude in timescales. The correlation of the transient shot with every other shot lies along the horizontal and vertical intercepts at n = 0 and grows in as time progresses, again exhibiting dynamics extending to ms timescales. Fig. 2B quantitatively compares the time-dependent integrated intensity on the detector with the time-dependent correlation as computed from Eqs. 1 and 2 in the ROI. As noted from a qualitative look (Fig. 1C), the integrated intensity, a probe of the volume fraction of the new phase growing in, initially grows quickly and exhibits minimal growth at longer times. The correlation function, however, grows in with a significantly more interesting temporal behavior, indicative of the more complex evolution of the heterogeneity associated with the new phase. The major feature is the long plateau that occurs between 500 ns and 100 µs, reflective of the lack of qualitative similarity in the details of the speckle pattern between transient and final frames in Fig. 1C. While the correlation approaches its final value after timescales of order 10 ms, there are still very long-time dynamics occurring out to ~100 ms, indicative of mesoscale dynamics of the phase transition that are not complete. This can also emphasized by the fractional change in the two-time correlation function $\Delta C/〈C(f,f)〉$ (Eq. 2) (Fig. 2B) which corrects for any spot-to-spot differences in speckle visibility. This fractional change follows a shape similar to that observed in the slow dynamics of heterogeneous soft-matter systems or supercooled liquids with a fast decay, long plateau, and then long-time fall-off (32, 33). The initial growth in C(t,f) (equivalently, the initial decrease in ∆C/C) corresponds to the emergence of the supercrystal satellite peak and its initial shift to its final position in Q space (SI Appendix, Fig. S4) for analysis of peak shape and location). Thus, although the dynamical response seems to be complete after a few μs as judged by standard crystallographic techniques, the correlation analysis reveals this to be incorrect. This conclusion was shown to be robust across selection of different ROI (SI Appendix, Fig. S5).

Fig. 2.

(A) Average two-time correlation plots at 500 ns (Top) and 3 ms (Bottom). The average was conducted across different sample locations pumped with the same pump–probe delay. (B) Comparison of the normalized integrated intensity (red) to evolution of correlation function $⟨C\left(t,f\right)⟩$ (black) (Top). Comparison of the normalized differential intensity (red) to differential correlation normalized to the final correlation value (black) (Bottom). Each time point was normalized to the average $C\left(f,f\right)$ at that time point. In both the Top and Bottom plots, the points after 3 ms were calculated by averaging selected n > 0 rows of the two-time plots and are included to show the asymptotic value of $⟨C\left(t,f\right)⟩$ and $⟨\mathrm{\Delta }C/C\left(f,f\right)⟩$. Insets show speckle correlation data at delays earlier than 1 µs.

The dynamics encoded in the correlation function are distinct from the evolution of properties of the peak other than intensity as well. The evolution of the transient peak can be seen in SI Appendix, Figs. S2 and S4. Selected transient images are shown in SI Appendix, Fig. S2 while the quantified peak width and peak center are shown in SI Appendix, Fig. S3. The evolution of the peak width (a measure of disruption to the long-range order of the system) and the peak center (a measure of the period of the emerging VSC phase) can both be seen to follow a faster pathway than the evolution of the correlation function, with no evidence for a plateau. Additional information about the collapse of the pristine phase mixture can be understood by measuring the decorrelation of the satellite peaks associated with the a₁/a₂ phase that is only present in the pristine state. The decorrelation is shown in SI Appendix, Fig. S6.

Discussion

To understand the microscopic processes responsible for the long-time nonequilibrium response, we first conducted a phase-field simulation of a PTO/STO superlattice as it transforms (34, 35) (Fig. 3). Diffraction patterns and associated correlations functions using Eq. 1 were computed for each simulated structure following reference (36) (Materials and Methods). First, the VSC phase nucleates at multiple locations and starts to expand, initiating the formation of the supercrystal phase with multiple grains. During this early time process, defects are generated at the boundary between the grains that do not anneal until much later in the simulation. Fig. 3A shows the real-space evolution from the simulation with additional movies in SI Appendix. The early-time process creates a local region of another intermediate structure with distinct periodicity from the original structure and the VSC phase as well as multiple sets of dislocation-like defects in the supercrystal region. This early process corresponds to an intensity increase of the supercrystal satellite peak that accounts for ~90% of its final equilibrium value but an increase in the correlation coefficient of the supercrystal satellite peak that accounts for only 50% of the final equilibrium value (Fig. 3B). The system then further undergoes a much longer process where the supercrystal defects gradually disappear. For the supercrystal satellite peak, ~10% of the final intensity and ~50% of the final correlation coefficient of the final value is gained within this second process. This gives rise to a two-component development of the correlation coefficient during the formation of the supercrystal (Fig. 3B), similar to that observed experimentally. As the process requires the transit and annihilation of defect pairs and grain boundaries, we expect the time duration to scale with the size of the simulation cell. Nevertheless, this shows the sensitivity of the correlation function to small-volume-fraction defects and domain boundaries as they annihilate and qualitatively capture the experimental results.

Fig. 3.

(A) Real-space phase-field simulation results of the heterogeneous evolution of the supercrystal growth as a function of time showing the z-component of the polarization vector in a 308 nm × 308 nm area at selected simulated timesteps labeled in the top left corner (Inset: corresponding computed diffraction pattern). (B) Comparison of the first supercrystal satellite peak intensity (purple) to correlation function (yellow) as defined by Eq. 2 from data in (A), over the entire first supercrystal satellite peak.

The phase-field simulations (Fig. 3) indicate that the different dynamics seen by the correlation function and the integrated intensity likely arise from the annihilation of defects at the boundaries of supercrystal-phase regions during the transformation. Qualitatively, such a sensitivity of the correlation function to such defect states and other types of heterogeneity can be understood from additional sensitivity in the correlation analysis to the diffuse scattering around the main Bragg peaks associated with nano/mesoscale ordering. An integration over an ROI in reciprocal space, as typically analyzed in a crystallographic experiment, integrates out the fine details of the diffuse scattering (37, 38) which encode disorder and heterogeneity associated with the growth of the new phase, and thus provides information only on the volume fraction. In contrast, the two-time correlation function sensitively probes this, and is sensitive to the fine details in the scattering that reflect disorder, heterogeneity, and fluctuations within the growth of the new structure. In particular, as the domain walls evolve it is the high wave vector (off-Bragg-peak) components which change and these are picked up by the correlation approach. Thus, this approach enables one to obtain direct information about domain walls, the local-strain fields at the boundaries of the bubbles of the new phase, and their dynamics (20, 22, 39–41).

To overcome the size limitation of the phase-field simulations, a series of phenomenological simulations were conducted, following approaches used in the modeling of spinodal decomposition (42). A grid of points, representing small-area elements undergoing a phase transition was established with each point assigned either a value of 0 or 1 corresponding to the starting phase or the final supercrystal phase, respectively. A two-dimensional Fourier transform of this grid enables a simulation of a transmission mode XPCS measurement. The dynamics of nucleation and growth were simulated in this model by creating small regions of the final phase at random locations and allowing them to grow as disks until they impinge on neighboring regions of the final phase, creating many local boundary regions (Materials and Methods). Fig. 4A shows the nucleation and growth process in one such simulation, together with the corresponding computed diffraction patterns. An animation of the growth process can be seen in Movies S2 and S3. As expected, as the local regions expand the diffuse scattering qualitatively contracts inward, encoding information about the mesoscale structures and boundary regions. This was verified by plotting the azimuthal average of the diffraction patterns for different timesteps (SI Appendix, Fig. S7).

Fig. 4.

(A) (Top row) Simulated nucleation and growth model snapshots showing isolated domains growing and impinging upon each other to form defect regions at the boundary between nuclei, followed by defect annihilation. (Bottom row) The corresponding diffraction pattern for each image in the Top row. Animations of these simulations can be found in Movies S2 and S3. (B) Average over 100 simulations of the correlation function between the transient and final computed speckle pattern in an annular region of interest. The generated defects are annihilated using two different waiting time distributions overlayed on the experimental raw correlation data. Inset is the computed integrated intensity with negligible difference for the two waiting time distributions (C) Overlayed average simulation and experimental normalized differential correlation data. Both (B) and (C) show the sensitivity in the long-time data to the waiting time distribution governing the annihilation of boundary defects.

Following an initial growth period, the boundary regions were allowed to reduce or annihilate at times sampled from a statistical waiting-time distribution (43) for the annihilation process, reflecting the stochastic jumps required to overcome an energy barrier associated with the merging/coalescence of domain walls. Similar models have been applied to domain–wall-based switching in ferroelectrics (44, 45) and to anomalous transport in disordered materials (46). In the simplest model, such distributions of waiting times arise from a heterogeneous distribution of energy barriers associated with these activated processes (47). The waiting times assigned to each boundary were sampled from an exponential distribution decaying with timescale t with $P\left(t;\tau \right)=\frac{1}{\tau }\exp \left(-\frac{t}{\tau }\right)$ and a “longtail” distribution with $P\left(t;\tau ,\alpha \right)=\frac{\alpha }{\tau }\frac{1}{{\left(1+\frac{t}{\tau }\right)}^{\alpha +1}}$ that decays at long times as $1/t^{\alpha +1}$. As a control, a uniform distribution was also considered. The results of using the uniform distribution are compared with other waiting time distributions in SI Appendix, Fig. S8. The exponential distribution is a prototypical memoryless distribution whereas the longtail distribution is associated with a scale-free fall-off and an infinite first moment (for $\alpha <1$) which gives rise to long-time, nonequilibrium dynamics, and corresponds to non-Markovian and subdiffusive behavior of the domain walls (46–48).

A comparison and fit to experimental data using two waiting time distributions is shown in Fig. 4 B and C. Comparing the experimentally extracted correlation and normalized differential correlation curves shows the greatest similarity between the long-tailed distribution case and the experimental data (Fig. 4 B and C). We obtain best agreement for $\alpha \approx 0.86,\tau ={1.4\times 10}^5$ in the case of the long-tailed distribution. In the case of the exponential distribution, the best agreement was found to with a $\tau \approx 4.1\times {10}^5$ (Materials and Methods for a description of the fitting procedure, and SI Appendix, Fig. S9 for a summary of the fit results).This simulation, despite the simplicity of its assumptions, thus captures the central experimental observations, showing sensitivity to the dynamics of the domain–wall boundaries as they dynamically evolve, encoded in the time-dependent correlation function. SI Appendix, Fig. S8 shows fits for different values of $\alpha$, showing the sensitivity of the correlation function to the power law exponent and defining a new approach for extracting this parameter. Fitting the correlation curves to simulations revealed that while using the exponential distribution is sufficient to produce a rapid rise in correlation of the speckle pattern, it is insufficient to reproduce the shape of the experimental curve at time delays greater than 10 ms. The best fit value for $\alpha$ indeed gives rise to dynamics with infinite first moment, underlying the long-lived nonequilibrium response observed here. This subdiffusive response may be understood in terms of the correlated interactions between different domains, as measured in prior slower timescale XPCS experiments probing jamming transitions (49, 50). We note that the time-scales observed in the correlation dynamics likely scale with the beam size of the probe X-rays, consistent with long time ergodicity breaking in the thermodynamic limit (5).

The extracted power-law-tail behavior of the waiting-time distribution for the domain–wall-annihilation process can in turn be related to an effective time-dependent growth of an average domain size, the central parameter within previously studied coarsening phenomena, phase-ordering kinetics, and other universal symmetry-breaking phenomena following an ultrafast quench (5, 22). We show in SI Appendix that in two dimensions, such a waiting-time distribution law gives rise to a domain-size growing algebraically in the scale-free manner ${L\left(t\right)\sim t}^{\frac{\alpha }{2}}=t^{0.43}$ in reasonable agreement with other theoretical work (5, 48, 51). We note prior theoretical modeling of percolation dynamics in nonequilibrium coarsening models has indicated the presence of two time-constants; a fast and a slow one, as observed here in the correlation function (52). We also observe that even simpler single nucleation models as described in SI Appendix can capture the observed delayed/two-step behavior of the correlation coefficient. These show enhanced sensitivity to the heterogeneous strain field at the boundary of the domain, consistent with the above models.

In summary, this work defines new approaches for visualizing and understanding the ultrafast, heterogeneous, and stochastic nonequilibrium dynamics of mesoscale defects, and more broadly, the nonequilibrium pathways materials follow as they transform. We show that the dynamics accessible in coherent scattering experiments are distinct from those accessible in other scattering experiments and encode processes extending to orders of magnitude longer times than previously thought. Future work may enable combination of these approaches within device geometries under applied fields, with important technological applications. For example, ferroelectric device switching is mediated by nucleation and growth processes similar to those probed here. Tracking the pathways of such transitions via correlative scattering approaches would enable new types of calorimetry including sensitive probes of dissipation and entropy growth, which in turn define the ultimate speed limits (53) and energy costs for switching (54). Expansion of X-ray free electron laser sources toward higher rep-rates should offer significant further improvements, potentially enabling direct real-space reconstruction of dynamically evolving defect pathways and their stochastic dynamics.

Materials and Methods

Experimental Setup.

X-ray photon correlation spectroscopy experiments were conducted at the XCS hutch at the Linac Coherent Light Source (LCLS) at the SLAC National Accelerator Laboratory. The X-ray photon energy used for the diffraction experiments was 9.5 keV. X-ray pulses were fired at sample positions at a repetition rate of 30 Hz determined by a pulse-picker. Diffraction patterns were recorded using epix100 detectors located 8 m away from the interaction point on the sample. Specifications can be found in ref. 55. The sample was pumped using a 50 fs 400 nm laser pulse produced by doubling an 800 nm laser pulse from a Ti:Sapphire amplifier.

Materials Growth.

The [(PbTiO₃)_n/(SrTiO₃)_n]_m (n— is the number of monolayers, n = 16; m— is the period of the superlattice, m = 8) superlattices were synthesized on single-crystalline DyScO₃ (110) substrates and via reflection high-energy electron diffraction (RHEED)-assisted pulsed-laser deposition (KrF laser). The PbTiO₃ and the top SrTiO₃ were grown at 610 °C in 100 mTorr oxygen pressure. For all materials, the laser fluence was 1.5 J/cm² with a repetition rate of 10 Hz. RHEED was used during the deposition to ensure the maintenance of a layer-by-layer growth mode for both the PbTiO₃ and SrTiO₃. The specular RHEED spot was used to monitor the RHEED oscillations. After deposition, the superlattices were annealed for 10 min in 50 Torr oxygen pressure to promote full oxidation and then cooled down to room temperature at that oxygen pressure.

Averaging Method for Speckle Analysis.

Each location on the probed sample produced a set of speckle patterns. One such set is shown in SI Appendix, Fig. S1, with the speckle pattern taken as the transient image marked. The LCLS has a certain amount of shot-to-shot variation in the spatial location of the X-ray probe pulses, leading to slight differences in location of the observed satellite peaks between collected frames. To ensure a comparable ROI is chosen in correlation analysis, a fitting procedure was done to find the center of the final satellite peak. First, noise was reduced in the images by setting any pixel value less than 0.25 times the value measured when a single photon is detected to 0. The average final satellite peak was calculated at each sample position by averaging the final satellite images taken at the location each normalized to the incident intensity measurement taken for that image. A Lorentzian peak shape with a linear background (Eq. S1) was fitted to the average vertical and horizontal profile to determine the center FWHM:

$$P\left(x;A,w,x_0,m,b\right)=\frac{A\ast w}{{(x-x_0)}^2+w^2}+m\ast x+b.$$ [S1]

Once a center was determined, an ROI is chosen both in the transient image and the final images relative to the fitted center. The correlation function described in Eq. 2 was then calculated over this ROI to produce the two-time plots shown in Fig. 2A. The average $C\left(t,f\right)$ value was calculated by averaging the correlation value between the transient image and the final images at a single sample location, then averaging across locations with transient images with the same sample delay. The error bars were calculated as the SEM correlation value across locations. The same procedure was used to calculate the change in peak center and peak width, shown in SI Appendix, Fig. S4 A and B. The peak width and position were extracted from the fitted Lorentzian peaks.

For the delays beyond 3 ms, the average correlation between the n > 0 frames was considered. The points shown in Fig. 2 were taken by averaging the correlation between each of the n $\in$ [1, 8] frames and the n $\in$ [9, 25] frames. The final point was taken by averaging the correlation between the n $\in$ [15, 24] frames and the n = 25 frame.

Multinuclear Phenomenological Modeling Procedure.

1,000 nuclei were generated on a 500 x 500 cell grid. The center locations of the nuclei were randomly chosen so that no two nuclei overlapped centers. The nuclei were all generated before any evolution to approximate the experimental behavior of nucleation occurring effectively instantaneously upon laser excitation. Each nucleus was assigned a unique number in the range [0,1,000). For each cell, the ratio

$$r=(d_2-d_1)/(d_2+d_1),$$ [S2]

was calculated, where d₁ is the distance between the cell and the closest nucleus, and d₂ is the distance between the cell and the 2nd closest nucleus center. Each cell (grid point) was assigned a unique index based on the two nearest nuclei for the purpose of assigning border regions to pairs of adjacent nuclei. The equation for generating the unique index is based on the Cantor pairing function $\pi \left(k_1,k_2\right).$

$$\begin{array}{c}m\left(n_1,n_2\right)=\pi \left(\left|n_1-n_2\right|,n_1+n_2\right)=\left(\left(n_1+n_2\right)+\left|n_1-n_2\right|\right)\hfill \\ \times \left(\left(n_1+n_2\right)+\left|n_1-n_2\right|+1\right)+\left(n_1+n_2\right),\hfill \end{array}$$ [S3]

where $n_1$ and $n_2$ are the indices of the individual nuclei and $m$ is the index assigned to the cell. To determine border regions, cells with r < 0.05 were permanently cell to 0. Initial nuclei were generated by creating regions between 1 and 3 cells in radius around the randomly generated centers. For 1,000 timesteps, a number of randomly chosen nuclei determined by sampling a Poisson distribution with mean of 63.2 at each timestep were allowed to grow 1 pixel in radius. The speckle pattern of the domain structure during this growth phase was calculated using the Fast Fourier Transform (FFT) at 51 evenly spaced timesteps between 1 and 1,001. The border regions were allowed to reduce or annihilate for 10⁹ timesteps. Each border region with a unique index m was assigned a waiting time sampled from a uniform distribution, an exponential distribution (Eq. 3a), or a long tail distribution (Eq. 3b). When the timestep of the simulation reached the waiting time of any given border region, that region was annealed by instantaneously changing to a region with effective r = 0.01. This reduced some border regions’ thickness and eliminated others. The initial border regions and the final border regions are shown in Fig. 4 A and B. The border defect regions are colored in Fig. 4 A and B according to the label m given to them according to Eq. S3. The speckle pattern during the defect reduction phase was calculated at 51 logarithmically spaced timesteps. All correlation functions were calculated for annular ROIs of inner radius 2 and outer radius 10. An example two-time of this system is shown in SI Appendix, Fig. S5. The correlation plot shown in Fig. 3 is the 102nd row (equivalently 102nd column).

Fitting Multinuclear Simulation Results.

To find appropriate values of the waiting time distribution parameters (e.g., $\tau$ for the exponential distribution and $\left(\alpha ,\tau \right)$ for the long tail distribution), a fitting procedure was performed. Each timestep in the simulation was taken to be 1 ns. For the exponential and long tail distribution, 50 simulations using the same procedure as above were performed except that the growth steps were not performed, and the growth was assumed to be completely done by 3 µs. The resulting $\mathrm{\Delta }C/C(f,f)$ curves were averaged. The $\mathrm{\Delta }C/C(f,f)$ value during the plateau was calculated by correlating the speckle pattern generated by the Fourier Transform of the inverted boundary image with threshold 0.05. The final state in real space was taken to be the inverse of the boundary image with threshold 0.01. See the previous section about the phenomenological simulations for a description of how the boundary images were generated. An example set of C and intensity curves for different values of $\alpha$ for $\tau =1.5\ast {10}^5$ are shown in SI Appendix, Fig. S6. The $\mathrm{\Delta }C/C(f,f)$ curve was normalized to decay from 1 by dividing the values by the $\mathrm{\Delta }C/C(f,f)$ value before any boundaries were annealed (e.g., the differential correlation between the boundary images) and compared to the experimental data that were normalized by dividing $\mathrm{\Delta }C/C(f,f)$ by the average $\mathrm{\Delta }C/C(f,f)$ in the time range [500 ns, 30 µs]. The residual between these two curves calculated at the experimental delay points was minimized using a nonlinear minimizer using the COBYLA algorithm until the values reported in the main text were found.

Phase-Field Simulation Method.

The simulation system of the PbTiO₃/SrTiO₃ superlattice contains 2 PbTiO₃ layers and 2 SrTiO₃ layers, where each layer has a thickness of 4.8 nm, equivalent to 12 unit cell lengths of the perovskite (PbTiO₃ or SrTiO₃) crystal lattice. The in-plane size of the system is chosen as $308\mathrm{nm}\times 308\mathrm{nm}$. The method and parameters for computing the scattering intensity of the structural domains are the same as performed by Yang et al. (31). An animation showing a 3D depiction of the $P_z$ as a function of timestep is shown in Movie S1.

Two simulations were performed with slightly different values of epitaxial strain. With a strain of 0.2%, an intermediate phase appears during the phase transition, but alternate pathways appear with different values of strain (SI Appendix, Fig. S12). Despite the difference in pathway, a two-step process nonetheless is apparent, indicating that the correlation function is still a sensitive measure of the defect formation and annihilation process which occurs generally in this formation process.

Mononuclear Diffraction Simulation.

A 3rd set of simulations further demonstrated the sensitivity of the correlation function to the dynamics and structure of a domain wall during a phase transition. In this model, shown in SI Appendix, Fig. S13, we assume a 2D periodic lattice of atoms and nucleate a new phase at the center of the grid, defined by a structure with slightly smaller lattice parameter. We imagine this new phase grows in by expanding radially with time until the old phase is subsumed by the new one. In the picture shown on the left of SI Appendix, Fig. S13A, we depict a snapshot of this transformation for the case of an atomically sharp domain wall separating the new phase from the old one. In contrast, the right figure of SI Appendix, Fig. S13A shows the corresponding snapshot for a diffuse domain wall where a Gaussian strain gradient separates the new phase from the old one. With these two charge density snapshots, one may then directly compute the corresponding diffraction patterns at each step in the simulation by Fourier-transforming the charge density with resulting snapshots shown in SI Appendix, Fig. S13B. Finally, from these diffraction patterns we may extract the time/radial dependence of the intensity and correlation function. For a sharp domain wall, the intensity and the correlation function track each other approximately, whereas for the more diffuse case, we find that the correlation function evolves significantly more slowly than the integrated intensity, showing as observed experimentally, that the correlation function more sensitively probes the structure of the domain wall.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

An animation of the evolution of the z component of polarization (parallel to the pseudo-cubic c axis) over the course of the phase field simulation presented in Fig. 3 of the main text.

Movie S2.

An animation of the real space simulation shown in Fig. 4 in the main text. Nucleated domains are allowed to grow until they impinge on neighboring domains, creating boundary regions that do not transform into the new phase and therefore do not satisfy the diffraction condition.

Movie S3.

An animation of the reciprocal space simulation shown in Fig. 4 in the main text. The central pixel dominates the simulated intensity signal, but the speckle patterns in rings around the central pixel represent the evolution of the growing phase.

Data, Materials, and Software Availability

Additional data and plots are included in the supporting information. All raw data and simulation code are available at https://zenodo.org/records/14532029 (56) and https://github.com/anudeepmangu/NucleationAndGrowthXPCS (57).