Intermittent Defect Fluctuations in Oxide Heterostructures

The heterogeneous nature, local presence, and dynamic evolution of defects typically govern the ionic and electronic properties of a wide variety of functional materials. While the last 50 years have seen considerable efforts into development of new methods to identify the nature of defects in complex materials, such as the perovskite oxides, very little is known about defect dynamics and their influence on the functionality of a material. Here, the discovery of the intermittent behavior of point defects (oxygen vacancies) in oxide heterostructures employing X‐ray photon correlation spectroscopy is reported. Local fluctuations between two ordered phases in strained SrCoOx with different degrees of stability of the oxygen vacancies are observed. Ab‐initio‐informed phase‐field modeling reveals that fluctuations between the competing ordered phases are modulated by the oxygen ion/vacancy interaction energy and epitaxial strain. The results demonstrate how defect dynamics, evidenced by measurement and modeling of their temporal fluctuations, give rise to stochastic properties that now can be fully characterized using coherent X‐rays, coupled for the first time to multiscale modeling in functional complex oxide heterostructures. The study and its findings open new avenues for engineering the dynamical response of functional materials used in neuromorphic and electrochemical applications.


Introduction
[3] This behavior, as characterized by metastable structures and abrupt transitions between configurations and states with similar energy levels, can be highly beneficial for emerging devices. [4]Algorithms exploiting uncertain states have already been developed to address complex problems such as the prediction of traffic patterns and protein function. [5]Here, the probabilistic (p)-bits fluctuate between 0 and 1, and probabilistic spin logic rather than Boolean logic is used to perform computations. [6]nterestingly, the stochastic behavior found in nanoscale materials and devices may differ considerably from those observed in larger systems due to fundamental differences in the statistical ensemble: [7] Such differences could be used to engineer the properties of p-bits and p-bit devices.The relevant experimental data in these fields are limited, however, due to the few number of techniques able to probe the stochastic behavior of nanoscale materials with sufficient resolution.X-ray photon correlation spectroscopy (XPCS), [8] the X-ray version of dynamic light scattering (DLS), leverages the laser-like qualities of synchrotron X-rays to unravel the dynamic pathways of nanoscale spatial orders in both soft and hard condensed matter under various in situ/in operando conditions, [9][10][11][12][13] by evaluating the temporal correlation of coherent X-ray scattering intensities.Recently, due to the many advances in X-ray sources [14][15][16] and the development of high-framerate detectors, [17][18][19][20] intermittent dynamics have been observed in a growing number of soft matter systems relaxing from farfrom-equilibrium states. [21,22]These intermittent dynamics are characterized by abrupt transitions between different metastable structures with similar energy levels, in contrast with the purely thermally driven fluctuations in ergodic systems (i.e., Brownian motion).Interest therefore arises as to whether intermittent dynamics exist in a complex oxide with competing phases differing in the oxygen-defect concentration, if the defect dynamics is stochastic or completely chaotic, and whether one can engineer the stochastic dynamics of such a system to control the dynamical response of functional devices.
We employ XPCS in the study of strained SrCoO x heterostructures.Epitaxial SrCoO x layers have been found to serve as a new type of random access memory element, with researchers show-ing that the local oxygen concentration can be used to controllably modulate conductivity, demonstrating its potential for memristive applications and neuromorphic computing. [23,24]We show here that the dynamic properties of SrCoO x can be tuned from stagnant to stochastic through epitaxial strain, as we demonstrate for ≈27 nm-thick layers under thermal equilibrium.Here, the pathway of such intermittent dynamics corresponds to a rapid mixing and de-mixing pattern, as revealed by simulations designed to describe hierarchical dynamics in hard condensed matter systems where microscopic driving forces lead to structural changes on the mesoscale.Our study provides a new avenue for understanding and tuning the stochastic dynamics expected to be critical for the operation of future nanoscale computing systems.

In Situ Measurement of Intermittent Defect Dynamics
When epitaxially strained to (LaAlO 3 ) 0.3 (Sr 2 TaAlO 6 ) 0.7 (LSAT) (001) substrates, the lattice constant of SrCoO x lies between two distinct phases: SrCoO 2.5 , an antiferromagnetic insulator with the brownmillerite (Bm) crystal structure, and SrCoO 3 , a ferromagnetic metal with the perovskite structure (Pv). [25,26]The microscopic structures of the two phases are closely related: the Bm structure is similar to that of the Pv but with oxygen vacancies lying along every other plane, resulting in alternating layers of oxygen octahedra and tetrahedra as depicted in Figure 1, while the Pv phase consists only of interconnected oxygen octahedra.
In previous work, we demonstrated that one can switch between the Bm and Pv phases by annealing in an inert environment versus an oxidizing environment at moderate temperatures. [25]The transition is topotactic and can be accomplished for single crystalline Bm and Pv without the loss of long-range order. [26]The reversibility of the transition was observed by monitoring the intensity of a half-order reflection such as the 00 1 2 , a Bragg peak that originates from the alternating layer structure in SrCoO 2.5 . [27]When oxygen is incorporated within the tetrahedral planes, the SrCoO 3 structure appears, eliminating the 00 1 2 reflection.The propagation of the Pv phase during oxidation is primarily 1D and along the surface-normal direction of the film, whereas the propagation of the Bm phase during reduction is 3D, as was shown in a recent study. [28]The system used to monitor the in situ behavior of SrCoO x in different environments is depicted in Figure 1.Due to the unique structure and the transition between Bm and Pv phases, it is worth noting that the measurement of nanoscale Bm/Pv domain boundaries in SrCoO x heterostructure also reflects the dynamic behavior of the oxygen vacancies in oxides.
We focus on equilibrium phase fluctuations observed within single-crystalline SrCoO x films.The experimental conditions are similar to those of the previous study [28] except that an inert environment is here maintained throughout the measurement.The sample was heated in a custom-designed furnace and encased within a poly-ether ether ketone (PEEK) dome under a constant flow of nitrogen at ambient pressure (Figure S1, Supporting Information).Figure 1 shows a region of reciprocal space for the sample under reducing conditions at 340 °C.The smooth surface and sharp film/substrate interface are made evident by the high contrast of the Kiessig fringes in Figure S2 (Supporting Information).The fringes were maintained at all temperatures.In addition, both the scattering intensity from the Pv/Bm phase and the oxidation state of the SrCoO x film were observed to remain stable under static conditions, as seen in Figures S3-S5 (Supporting Information).
After thin film synthesis, strontium cobaltite is in the brownmillerite phase and remains in this phase in non-oxidizing environments.It should be noted, however, that while the 00 1 2 reflection is present, the average film composition is not necessarily SrCoO 2.5 .We determined the oxygen concentration by performing X-ray absorption near-edge spectroscopy (XANES) at the Co K-edge (Figure S5, Supporting Information).At 340 °C in a nitrogen environment, for example, the film stoichiometry is SrCoO 2.6 .
The measurements were performed adjacent to the 001/2 peak, with the diffractometer aligned at the first Kiessig fringe on the lower angle end (Figure S2, Supporting Information), consistent with our previous study. [28]While the integrated intensity is stable, the intensities of the individual speckles do vary with time, where the counting time of each speckle pattern was short enough to be representative of the static arrangement of scatterers.The dynamics can be described by two-time correlation maps, where the correlation function is given by ref. [29]   C where Here, 〈…〉 Q indicates an average over the entire scattering region on the detector image to improve the statistics of the correlation function, [30] and Q corresponds to the scattering vector for the reflection of interest.The incoherent scattering background, I inc (Q), is equivalent to the X-ray micro-diffraction pattern acquired with an incoherent X-ray beam, for example, that from a rotating anode source or a bending magnet synchrotron beamline.I inc (Q) then corresponds to the spatiotemporal average of the scatterers within the X-ray beam and is therefore timeinvariant for the Bm phase under thermal equilibrium.The values were obtained by passing the coherent X-ray scattering intensities through a Savitzky-Golay smoother to filter out the highspatial-frequency speckles, as in previous studies. [30]wo-time correlation maps are shown in Figure 2 for the 00 1 2 position at different temperatures, and sharp changes in color contrast demonstrate the occurrence of significant fluctuations.For example, when t 1 = t 2 = 600 s at 340 °C (Figure 2e), the correlation is high (≈0.18),as the speckle pattern is correlated with itself.However, C 2 drops to nearly zero at t 1 = 600 s and t 2 = 910 s, returning to a high value at t 2 = 950 s.Since the intensity originates from the 00 1 2 reflection, this implies that the spatial organization of oxygen vacancies at 600 s is very different from the one at 910 s, while the configurations at 600 and 950 s are similar.On average, the oxygen vacancy structure appears to decorrelate and recorrelate every ≈100 s in this temperature regime, although the behavior is non-periodic.In addition, the oxygen vacancies appear to be more dynamic as temperature increases, an observation suggested by the qualitative trend that the overall sizes of the correlated regions decrease and the differences between the correlation coefficient of the correlated and decorrelated regions increase at higher temperatures (Figure 2a-e).The implication is that although the Bm phase is thermodynamically stable as per the 00 1 2 integrated intensity, the tetrahedral structure/oxygen vacancy structure is highly dynamic.This is in sharp contrast with the behavior of the Bm phase for SrCoO x /SrTiO 3 (STO) at 340 °C (Figure 2f), where the same underlying tetrahedral structure/oxygen vacancy structure appears stable at 340 °C and remains so from 300 to 350°C in the N 2 environment. [28]e show the range of in-plane lattice parameters for SrCoO 2.5 , SrCoO 3 , LSAT, and SrTiO 3 at room temperature in Figure 2g.As seen, the SrCoO 2.5 lattice constant matches that of STO while that of LSAT lies between the Bm and Pv phases.
We further note that the progressive increase of the amplitude and frequency of the fluctuations with temperature for SrCoO x /LSAT suggests that the equilibrium dynamics of the Bm phase is thermally driven, as was seen for charge ordering domains in other epitaxial systems. [31,32]We will later show that the energy barrier for spatial reconfiguration of Bm phase is E A = 0.74 eV.This corresponds to a factor of three difference in the Boltzmann factor exp(−E A /kT) between T = 300 and 350 °C.The thermally driven equilibrium dynamics time scale is therefore expected to be three times faster at 350 °C compared to 300 °C, which is qualitatively consistent with the observation in Figure 2. Quantitative evaluation of the temperature-dependent behaviors, however, is not yet feasible due to the large dispersion of the fluctuation time scales.This is particularly true for the longest time scale that is comparable to the stability time of the experimental setup.Future studies will entail longer measurements at XPCS beamlines with greater spatial beam stability, with beam position monitoring and the ability to correct long-term beam drifts. [33,34]To obtain a detailed understanding of the underlying microscale defect dynamic processes that lead to the observed strain-controlled fluctuations measured by XPCS, we developed a density-functional theory (DFT) informed phase-field model, as detailed in the next section, for the Bm phase under the measured strain conditions and oxygen chemical potentials and compute C 2 to directly compare with experiments.

Defect Dynamics Modeling
Bakken et al. [35,36] have shown that oxygen vacancy-vacancy interactions can be considerable in perovskites such as SrCoO x , leading to preferred configurations such as the brownmillerite structure in vacancy-rich regions.We provide a semi-quantitative description of the defect behavior, combining phase field modeling with ab initio calculations.As described in Supporting Information, energy parameters from the latter were used to inform the phase field model.Particularly, the mobility of vacancies and oxygen ions were derived from the energies obtained using ab initio calculations for the different substrates used in experiment.Energies from four different defect configurations are shown in Figure 3a for SrCoO x strained to LSAT, or for comparison to SrTiO 3 , with the structures depicted below.Additionally, we indicated the path of the oxygen ion hopping to transition between the configurations, traveling from the Pv to Bm phase.The empty blue-gray circles in the Figure 3a show the newly formed vacancy (in the Pv phase) and the vacancy where the oxygen ion eventually settles within the Bm phase.[39] The local free energy can then be described by three independent terms: a chemical term, g ch (,Ω), an elastic strain term, g el (), and an interfacial term with , the gradient energy as per the Cahn-Hilliard formalism. [40]The total free energy of the system is the sum of these contributions integrated over the entire volume: The oxygen stoichiometry in the Bm, Pv, and mixed state defines the value of the order parameter where c is the local concentration of oxygen ions, c Bm = 0.028 mol cm −3 is the minimum concentration of oxygen in the brownmillerite, and c Pv = 0.088 mol cm −3 is the maximum concentration of oxygen in the perovskite.In terms of oxygen ions and vacancies,  represents the local oxygen concentration while 1 −  is the local vacancy concentration.The temporal evolution of  then takes the form of the Cahn-Hilliard equation: [40] (x, t) where is the diffusion tensor, and  is a stochastic component that corresponds to the local thermal fluctuations in the field. [41]The parameter Ω describes oxygen ion-vacancy interactions as in the regular solution model.More details are provided in Supporting Information.The assumption of a binary mixture with the conservative dynamics equation is a simplification that allows us to build a phase field model and qualitatively describe the dynamics in complex oxide films.To build a fully quantitative model, complete thermophysical information for the system is needed which is not currently available.However, as we demonstrate below, this approach allows one to correlate fluctuations found in the C 2 map with the organization of the locally separated ion and vacancy phases.We performed phase field modeling with  = 0.5, with the substrate lattice parameter set to either that of bulk LSAT or SrTiO 3 and calculated the resulting two-time correlation function.The effect of epitaxial strain on the interfacial energy between the Bm and Pv phases was indeed found to be significant, with the Bm-Pv interfaces being much more stable for films grown on LSAT rather than SrTiO 3 ; the energy barrier for oxygen vacancies to diffuse across the interface was also found to be much smaller.Figure 3b shows that the free energy curve (black) exhibits a double well for SrCoO x /LSAT, with a very shallow energy barrier.For SrCoO x /SrTiO 3 (Figure 3c), the minimum is shifted significantly toward the configuration favoring the Bm-phase assuming all other parameters unchanged.Figure 3d shows C 2 for Ω = 0.104 eV and the LSAT substrate, in which case the free energy has a single well.The homogeneous pattern corresponds to the real space simulation images exhibiting a completely mixed phase (Figure S6a, Supporting Information).The results in Figure 3e,f shows C 2 maps for Ω = 0.124 eV for SrCoO x /LSAT and SrCoO x /SrTiO 3 , respectively.In the case of the former, the choice of a higher ion-vacancy interaction parameter results in a double well potential, which then leads to definitive phase separation and the emergence of stable Bm-Pv phase boundaries (Figure S6b, Supporting Information).This is in sharp contrast to results for the SrCoO x /SrTiO 3 heterostructure (Figure 3f), where the potential is shifted to the Bm phase due to the energy of the lattice mismatch.This results in the absence of phase separation and a C 2 similar to the Figure 3d with no features observed.Furthermore, we find that to reproduce the behaviors seen in Figure 2, it is necessary to make Ω time dependent, as per the Langevin equation where  is a width of the distribution as shown in the inset of Figure 3c (details in Supporting Information).This allows the immediate appearance of dynamic fluctuations in the two-time correlation function depending on the value of Ω, as demonstrated in Figure 4a.Here, the separation of phases is followed by an uncorrelated regime where the interfaces between the phases disappear due to thermal noise and interdiffusion.We compare this C 2 function map to ones obtained from experiment (e.g., Figure 2e).
Similar to the experimental results (Figure 2f), the C 2 function map calculated for SrCoO x strained to the SrTiO 3 substrate (Figure 4b) shows minimal fluctuations, particularly when compared to those for SrCoO x /LSAT, even when the values of Ω and  are identical.Although the correspondence between the experimental and calculated maps is rather qualitative, our simple model demonstrates directly the critical role of epitaxial strain on dynamic properties in the SrCoO x system.
The thick, red profile in Figure 3b that represents ±0.004 eV fluctuations in Ω about an average value (0.119 eV) is characteristic of the free energy function responsible for the dynamic fluctuations observed for SrCoO x /LSAT.The real space patterns that represent the Bm and Pv phases within the correlated region shown in Figure 4a is exhibited in Figure 4d for t = 930 s and Figure 4e for t = 945 s.The disappearance of the Bm/Pv interfaces happens in the case of fully mixed oxygen ions and oxygen vacancies, as shown in Figure 4c at t = 745 s.Similar changes in Ω do not result in fluctuations for SrCoO x /SrTiO 3 heterostructures.The calculated C 2 , shown in Figure 4b, appears to be mostly timeindependent, as in the experiment (Figure 2f).
Order parameter profiles along the white line indicated in Figure 4c-e demonstrate the separation of oxygen ions and vacancies into two phases resulting in appearance of interfaces (Figure 4f-h).The presence of the local thermal noise  in Equation 5 results in the nonuniform profile shown in Figure 4f.However, more pronounced minima and maxima in the order parameter, , appear due to the formation of ordered oxygen ion and oxygen vacancy phases (Figure 4g,h).The spatial and temporal variations in  can then qualitatively describe the experimental C 2 maps shown in Figure 2a-e.
While the activation energy of 0.74 eV calculated from DFT is similar to one of the configurational energies in Figure 3a, these energies are distinct from the Ω chosen to maintain balance between the equilibrium mixed state and nonequilibrium phase separation to produce fluctuations in the calculated C 2 .There are many possible reasons for this including the absence of full thermodynamic information for these systems, but we note that different correlations may exist for different time and size scales.Additionally, thermodynamic data on ion-vacancy interaction, mobility in the solid matrix, and composition-dependent properties are necessary to fully understand the ion-vacancy, liquid-like dynamics within the matrix of complex solid oxides.With the advent of more coherent light sources, it will be possible to better investigate the effects of system size on dynamic behavior.

Conclusion
Defects such as oxygen vacancies are acknowledged to be key to many of the correlated, multifunctional, and neuromorphic properties discovered in nanoscale complex oxide heterostructures (e.g., ref. [42-47]).Given the recent discovery of superconductivity in highly reduced and vacancy-ordered systems, [48][49][50] understanding their dynamic behavior in a variety of conditions has become more urgent, particularly as mixtures of ordered phases are likely.The current results demonstrate how coherent X-rays can be used to investigate vacancy dynamics in such heterostructures, and we find that coherency strain is critical to the vacancy structure as well as to the dynamic behavior.For SrCoO x /LSAT at elevated temperatures, fluctuations between very different defect configurations are observed with relatively slow dynamics.This can be captured by a Cahn-Hilliard-type equation, treating the concentration of oxygen ions and oxygen vacancies as a two-component binary mixture which defines the brownmillerite and the perovskite phases.A Ginzburg-Landau free energy that drives the dynamics, that is, the mixing of the two phases, is developed using parameters derived from ab initio calculations.The results suggest that fluctuations between the Pv-Bm phases depend on variations in the binding of oxygen-vacancy pairs, as well as to epitaxial strain, both of which differ significantly for films grown on either the LSAT and SrTiO 3 substrates.This discovery and the gained mechanistic insight into the intermittent dynamics of oxygen vacancies in strained heterostructures can be also informative to further elucidate the relaxation of glassy states that contain internal strains, [51,52] suggesting intermittent dynamics as a key subject for a broad range of studies as well as providing novel ways of controlling the dynamical response of various functional systems, such as in neuromorphic devices.

Statistical Analysis
XPCS analysis from the CCD detector images acquired at 8-ID-E is performed using customized Matlab scripts consistent with the approach detailed in previous studies: [13,30,53] 1.A series of 20 detector images were acquired with no X-rays (i.e., dark images) prior to each XPCS measurement; 2. The average of the dark images is subtracted as background from all detector images; 3.After the background subtraction, Readout from pixels lower than 4 times the standard deviation calculated from the 20 dark images is rejected and set to 0; 4. The incoherent scattering background, which corresponds to the X-ray scattering pattern generated by an incoherent Xray source (e.g., a rotating anode), is calculated by digitally smearing out the speckles in the coherent scattering patterns.In this study, the incoherent scattering background is calculated by passing the time-averaged detector images through a Savitzky-Golay high-frequency filter.The time-average is for improving the statistics and is only possible because the sample is at equilibrium and the overall scattering intensity does not vary with time, as shown in Figure S3 (Supporting Information); 5.The normalized intensity fluctuation is calculated by subtracting the speckles from Step 3 with the incoherent scattering background from Step 4, and then normalize pixel-by-pixel with the incoherent scattering background; 6.The two-time correlation between every pair of detector images is calculated from the pair-wise correlation of the normalized intensity fluctuation extracted from each detector image.The correlation is then averaged over a 150×100 pixel regionof-interest (ROI) marked by the white rectangle in Figure 1 to improve the statistics.

Figure 1 .
Figure 1.Schematics of the in situ XPCS experiment and speckle pattern evolution.The top left shows the oxygen octahedra (yellow) representative of the perovskite (Pv) SrCoO 3 phase and the alternating octahedral (yellow) and tetrahedral (purple) layers representative of the brownmillerite (Bm) SrCoO 2.5 phase.The fluctuations of the Pv and Bm domains (yellow and purple regions in the images below) lead to the temporal decorrelations seen in the intensity speckle patterns (detector images), and the pair-wise correlations between the speckle patterns acquired at different experimental times (two-time correlation, top-right) provide a quantitative measure of the fluctuating nature of the system.The white rectangle on the detector image indicates the region-of-interest (ROI) for the Q-average in Equation (1).

Figure 2 .
Figure 2. Two-time correlation C 2 measured near the 00 1 2 reflection of SrCoO x .Results for SrCoO x /LSAT are shown in (a-e) for 300-340 °C and (f) for SrCoO x /STO at 340 °C.The color scale corresponds to the values of normalized correlation coefficient.g) Range of the in-plane lattice constant illustrating the bulk values of SrCoO x and the LSAT and STO substrates.The colors reflect those of the different phases in Figure 1.

Figure 3 .
Figure 3. Summary of the ab initio and phase field results.a) Calculated energies for different oxygen vacancy configurations in the SrCoO x structure.The configurations are depicted below with arrows indicating the paths for oxygen ion diffusion during the transition from configuration A to B. The horizontal dotted line indicates the hypothetical interface between the perovskite (above) and brownmillerite (below).b,c) Normalized, homogeneous part of the free energy density and the chemical and elastic components are shown as a function of the oxygen vacancy concentration, , for SrCoO x films coherently strained to LSAT (b) and SrTiO 3 (c) (001).The four energy profiles demonstrate the effect of different terms in Equation (4).A distribution in the oxygen ion-vacancy interaction parameter, Ω, as shown in the inset of (c) leads to the red profiles in (b,c) where  is the width of the distribution.d-f) Calculated C 2 maps for different substrates and fixed values of Ω.

Figure 4 .
Figure 4. Two-time correlation C 2 maps and real space images determined from simulation.a,b) Calculated C 2 maps for different substrates and the interaction parameter Ω = 0.119 ± 0.004 eV.c-e) Real space simulation snapshots of phase fluctuations for SrCoO x on LSAT substrate at times (745, 930, and 945 s) that correspond to the correlated and decorrelated domains in C 2 .The horizontal white lines correspond to 7.5 m.f-h) Profiles of  (ion concentration) along the white lines marked in (c-e).As observed, the ordered vacancy domains have a correlation length on the order of a few micrometers.
work was led by the Center for Nanophase Materials Sciences (CNMS), which is a US Department of Energy, Office of Science User Facility at Oak Ridge National Laboratory and partly supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division.Work on the sample was supported by US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division.The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory ("Argonne").Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357.The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.http://energy.gov/downloads/doe-public-access-plan.