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Learning from Imperfections: Predicting Structure and Thermodynamics from Atomic Imaging of Fluctuations Lukas Vlcek,†,§,∥ Maxim Ziatdinov,‡,§ Artem Maksov,‡,§,⊥ Alexander Tselev,# Arthur P. Baddorf,‡,§ Sergei V. Kalinin,*,‡,§ and Rama K. Vasudevan*,‡,§ Downloaded via UNIV OF EDINBURGH on January 28, 2019 at 07:40:15 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
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Materials Sciences and Technology Division, ‡Center for Nanophase Materials Sciences, and §Institute for Functional Imaging of Materials, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States ∥ Joint Institute for Computational Sciences, University of Tennessee, Knoxville, Tennessee 37996, United States ⊥ UT Bredesen Center for Interdisciplinary Research, University of Tennessee, Knoxville, Tennessee 37996, United States # Department of Physics, CICECO - Aveiro Institute of Materials, 3810-193 Aveiro, Portugal S Supporting Information *
ABSTRACT: In materials characterization, traditionally a single experimental sample is used to derive information about a single point in the composition space, while the imperfections, impurities, and stochastic details of material structure are deemed irrelevant or complicating factors in the analysis. Here we demonstrate that atomic-scale studies of a single nominal composition can provide information about microstructures and thermodynamic response over a finite area of chemical space. Using the principles of statistical inference, we develop a framework for incorporating structural fluctuations into statistical mechanical models and use it to solve the inverse problem of deriving effective interatomic interactions responsible for elemental segregation in a La5/8Ca3/8MnO3 thin film. The results are further analyzed by a variational autoencoder to detect anomalous behavior in the composition phase diagram. This study provides a framework for creating generative models from a combination of multiple experimental data and provides direct insight into the driving forces for cation segregation in manganites. KEYWORDS: scanning tunneling microscopy, statistical inference, generative model, segregation, manganite, thin film
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experimental side, high-throughput combinatorial approaches have seen substantial successes in the recent past.8−11 At the same time, theory-guided materials discovery has experienced fast growth fueled by the available computational resources and development of advanced machine-learning (ML) algorithms. For instance, first-principles calculations are now routinely performed to narrow down the potential list of candidate materials with desired properties.12,13 These systematic advances enabled the rise of libraries such as the Materials Project, AFLOWLIB,14 JARVIS-DFT,15 Polymer Genome,16 and many others. Most recently, several ML approaches have been proposed to predict material properties.17−19 In general, these require sufficient amounts of training data that would allow the algorithms to infer and generalize relations between composition and observable properties. These predictions can be supplemented with additional experimental results within a
ccelerating materials discovery is critical to solving many of the world’s challenges, as can be reflected by large-scale endeavors such as the materials genome initiative.1−3 The traditional materials discovery workflow relies on the preparation and characterization of individual samples, effectively resulting in information about a single point in the chemical space per sample synthesized. Nonidealities such as chemical segregation, point and extended defects, impurity phases, and the like are traditionally perceived as limitations that negatively affect the veracity of the data, and synthesis strategies are traditionally aimed at avoiding these.4 The inherent atomic disorder, such as the stochastic distribution of atomic configurations, is often treated as selfaveraging and irrelevant to macroscopic properties. However, for materials such as ferroelectric relaxors, charge-ordered manganites, or filamentary superconductors, nanometer-scale heterogeneities are defining features critical to understanding the physical functionality and still remain one of the hot topics in condensed matter physics.5−7 In the modern era, there has been sustained effort to alter this paradigm and accelerate materials discovery. On the © 2019 American Chemical Society
Received: October 19, 2018 Accepted: January 4, 2019 Published: January 4, 2019 718
DOI: 10.1021/acsnano.8b07980 ACS Nano 2019, 13, 718−727
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Cite This: ACS Nano 2019, 13, 718−727
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ACS Nano Bayesian framework for experimental design optimization,20−22 to reduce the number of experiments taken to arrive at the optimal material given a set of input (processing) parameters. The main challenges in this area of research remain to be the limited amount of useful information extracted from experiments, poor interpretability of many ML models, and limitation only to predicting properties that were directly targeted in model training. A straightforward path to overcoming these drawbacks is to incorporate the available experimental information into a suitable and physically plausible generative model. Here we pose that by capturing the characteristic structural and chemical fluctuations that exist within a single chemical composition, we can infer the relevant interactions and produce a generative model that can predict properties over a range of scales in a finite region of the chemical and temperature space. In this approach, we combine the methods of statistical hypothesis testing with the principles of statistical physics, in which macroscopic system behavior can be understood as resulting from the interactions between its constituent components, and these interactions allow us to predict the system behavior as a function of thermodynamic variables. Similar ideas of analyzing observed fluctuations have been used in cases such as biomolecule unfolding via atomic force microscopy,23,24 or more recently in the study of the dynamic trajectories of DNA molecules in a nanofluidic environment, in which the measurements of stochastic trajectories were converted to equilibrium free energies.25 In this work, we aim to use the compositional and structural fluctuations in the quenched (static) system to build a generative model encoding the effective interactions in the system. Although feasible in principle, in practice, determining the form and parameters of generative models to match experiment is a difficult task. For instance, bulk property measurements on many samples can be carried out on a grid across the temperature−composition space, with the parameters of the generative model adjusted to match this data. However, if imaging modalities containing information about complex correlations between the system components are available, an alternative approach becomes practical. In this case even a single image can provide a large number of meaningful constraints on the model’s parameters in the form of statistics of atomic configurations, and the resulting optimized model can be used to predict phase diagrams and microstructures over a wide range of temperatures and compositions. Here, we demonstrate this approach for a model system, focusing on elemental segregation in a La1−xCaxMnO3 thin film, a material which is relevant for oxide fuel cell applications and spintronics and which has been studied previously by both experimental26−28 and combined experimental-theoretical29 approaches. We show that experimental data from scanning tunneling microscopy (STM) images can be consistently combined with electron spectroscopy results (providing a chemical profile as a function of depth) on thin-film La5/8Ca3/8MnO3 to infer the parameters of a three-dimensional (3D) generative statistical mechanical model that provides insights into subsurface microstructure and the thermodynamics of elemental segregation. The generated structures across the composition and temperature space are subsequently processed by a variational autoencoder approach to detect possible anomalies within this space.30,31 We find that the model of effective metal−metal interactions incorporating
the two independent sources of experimental information leads to the predictions of weak segregation forces in bulk (attributable to elastic effects) and weak desegregation forces in the surface (attributable to electrostatic interactions), in agreement with quantum-chemical calculations.29 In addition, we quantify the influence of surface oxygen on elemental segregation, which has not been reported before. The results demonstrate that leveraging recent major advances in imaging and ML in combination with the fundamental concepts of statistical mechanics provides opportunities for understanding physics from atomically resolved imaging and can lead to rational design of experiments and accelerated materials discovery.
RESULTS AND DISCUSSION General Approach. In the first step of the presented method, images from STM are analyzed to extract surface atomic configurations. This procedure is aided by available complementary constraints, such as the total stoichiometry, concentration profiles from X-ray photoelectron spectroscopy (XPS), or first-principles calculations,32 which reduce the atom classification error resulting from the inherently noisy electron or STM images. After the positions and types of each atom are identified, the histograms of representative local configurations are collected. The configurations are chosen to contain information about the direct correlations between atoms,33 roughly spanning the range of direct atom−atom interactions. In the present case, configurations containing the nearest and next-nearest metal atom neighbors are included. The statistics of such configurations represents a natural signature, or fingerprint, of the observed structure. We note that this approach to feature selection is a specific case of the bag-of-visual-words ML method used for image classification.34 A generative model is then optimized by minimizing the statistical distance metric between the target and predicted histograms,35−37 selecting a model whose predictions are the most difficult to distinguish from the experimental data using rigorous statistical hypothesis testing. The actual search over the model parameter space to minimize the statistical distance loss function is accomplished using the perturbation technique,38,39 which drastically reduces the computational cost of the inverse problem solution. The resulting model can be then used to predict the system’s response to changing thermodynamic conditions, such as temperature or chemical potential, and optimize these conditions to create microstructures associated with desired materials properties. The generated 3D structures can be further processed by unsupervised learning to discover new features by variational autoencoders, which have been shown to be useful in delineating phase transitions in both static and dynamic models in the recent past.31,40 Elemental Segregation in Thin-Film Perovskite from Microscopic Imaging and Spectroscopy. Information about the distribution of La and Ca atoms in the surface layer of perovskite La5/8Ca3/8MnO3 thin film was extracted from in situ STM images, informed by density functional theory (DFT) calculations.41 In the present case, the film was grown by pulsed laser deposition on (001) etched SrTiO3 substrates and is of thickness ∼50 unit cells (see Methods section). We have previously explored the growth, surface reconstructions, and terminations of these films through a combination of in situ atomic-scale imaging, low-energy 719
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Figure 1. In situ STM image and analysis of the A-site terminated surface of La5/8Ca3/8MnO3 thin film. (a) STM image (scale, 5 nm) of the Asite terminated surface showing a (√2 × √2) R45° reconstruction. A unit cell is boxed in blue. The FFT of the image is shown in (b) from which the lattice parameter can be calculated, which confirms that the lattice parameter is ∼0.56 nm, which is ≈√2a, where a is the lattice constant of the SrTiO3 substrate (a = 0.3905 nm). (c) Schematic of the (√2 × √2) R45° A-site terminated surface, which is formed by removing a single O atom from every other cell from the original (1 × 1) structure. (d) Image in (a) after filtering with PCA sliding window denoising algorithm. (e) Expanded view of region in (d), with simple thresholding to better locate the atom centers and lobes of intensity. The bright atoms are the oxygens, and lobes of intensity (elongations) around each O can sometimes be seen. If the intensity within these lobes is above a certain threshold, a La atom is placed; otherwise, a Ca atom is presumed present. (f) Fully analyzed image showing the O sublattice, along with the La and Ca. Note that the image captures an antiphase domain boundary with oxygens in the two respective domains colored red and blue.
Figure 2. Structural statistics of LCMO thin film from experiments (yellow) and simulations (blue). The histograms of local configurations for metal atoms surrounding a surface oxygen (a) and for configurations without a surface oxygen (b). The concentration profile of Ca atoms across six top layers (c). The snapshots of atomic configurations in the top (d) and bulk (e) layers from MC simulations of the LCMO thin film. La3+, black; Ca2+, yellow; surface O, blue.
√2)R45° lattice, with every second unit cell missing an apical oxygen atom. Because these oxygens contain the highest density of states at the Fermi Level, they are highly visible in
electron diffraction, and angle-resolved XPS, as published elsewhere.42,43 As shown in Figure 1a, the surface of the (La,Ca)-terminated film is reconstructed into a (√2 × 720
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ACS Nano STM images.42 The raw images were cleaned using the principal component analysis-based (PCA) filtering, and the position of the oxygen atoms was found using a motifmatching approach.44 This algorithm is freely available through the open-source pycroscopy package.45 Interestingly, both in raw and cleaned images, a fine structure can be seen around some of the oxygen atoms. Specifically, at some sites, the electronic density (intensity) is shifted somewhat away from the presumptive lattice site and appear to be elongated along the ⟨110⟩ directions. These elongations are in the directions of presumed metal (La, Ca) atoms, as seen in Figure 1d,e (these are also visible in raw images, see Figure S1). This observation is consistent with the results of our DFT calculations of charge density (as well as with basic chemical considerations),46 which predicted a higher electron density located between oxygen and La3+ as compared to Ca2+. Therefore, we used the DFT-based insights to associate these lobes of higher intensity with the presence of neighboring La3+. This information along with the known relative La/Ca surface concentration determined from XPS experiments allowed us to assign the positions of La and Ca in the surface layer, as shown in Figures 1f and S3d−f. Note that the Figure 1f captures an antiphase domain boundary with oxygens in the two respective domains colored red and blue. It can be seen that many of the La and O atoms form structures resembling a self-avoiding maze of paths, rather than segregated clusters or being purely randomly distributed. These patterns indicate specific interactions between the surface species that depend on the presence of the apical O atoms. To capture the effect of O atoms on elemental segregation for the use in subsequent model optimization, we collected the histograms of local configurations separately for those with and without the presence of apical O, as shown in Figure 2a,b, respectively. In these figures, the histograms obtained from the experimental images are shown as yellow bars, with the corresponding configurations of La and Ca particles plotted above. The higher frequency of La-rich configurations in the presence of oxygen in Figure 2a relative to Figure 2b indicates stronger affinity of O atoms to La compared to Ca. Note that the histograms contain error bars (1 standard deviation) obtained by undertaking the same image analysis for three independent images shown in Figures S2−S4. While the STM images provide detailed information about the atomic correlations in the top layer, they may not sufficiently constrain the elemental distribution deeper within the thin film, especially if the top layer can be expected to have distinctive properties and composition. To reduce this uncertainty, we complemented the imaging data with angleresolved XPS measurements that allowed us to infer the concentration profile across the thin-film layers.42 The data showed that for this particular film, the surface concentration was ∼ La0.5Ca0.5MnO3 and that the bulk concentration is recovered in the deeper layers. Quantitative analysis of angleresolved XPS data is difficult due to large number of unknown parameters in modeling, but experiments on LCMO show that the segregation is likely limited to only two to three unit cells at the surface, as in the study by Plummer et al.26 Therefore, the concentration profile was assigned as varying exponentially from x = 0.54 to x = 0.375 three unit cells into the film, corresponding roughly to the correlation length of 0.5 unit cells as found in ref 26.
The Model of Thin-Layer LCMO. The effective model of LCMO was designed with the goal of simplicity to prevent overfitting in the presence of statistical noise. Therefore, we have represented the distribution of La3+ and Ca2+ particles within the thin layer by a lattice model and treated the Mn and O atoms as rigid scaffolding. Arguably, the Mn oxidation state (III vs IV) can have an effect on the distribution of La and Ca atoms. However, as shown below, the present simple effective model is able to reproduce the available experimental data, which means that its further refinement would require additional independent information to prevent overfitting. We should also note that, to a certain degree, information about the Mn valence may be effectively captured by the presence or absence of apical O atoms, which are strongly correlated to Mn oxidation state, and whose interactions are included. The type of particle interactions was chosen based on prior insights made by others as well as our own studies. Lee and co-workers identified and analyzed two main contributions to elemental segregation in perovskite manganites, elastic and electrostatic, and argued that the former will vanish within the surface layer,29 while other studies have detected metallic character of the top layer46 for a particular LCMO surface,47 which may uniquely influence the interactions among the toplayer atoms. For this reason, we have retained the flexibility to adjust the effective interactions of bulk and surface atoms independently. Also, the fact that the XPS data42 show clear segregation between bulk and the top layer, which changes as the thin film grows, suggests that the metal atoms are mobile and able to adjust to the different thermodynamic gradients in bulk and surface. Therefore, we will assume a statistical mechanical model of an equilibrium system. For interactions between bulk Ca and La atoms, we only consider a single effective parameter wCL that controls the segregation reaction (Ca − Ca) + (La − La) ↔ 2(Ca − La). At the surface, we include specific interactions between pairs of metal atoms as well as explicit interactions of metal atoms with apical O. Taking these considerations into account, we can write the energy ui of configuration i of the thin-layer system as ui = wCL ∑ δib , CL + wLL ∑ δis , LL + wLO ∑ δis , LO {C , L} 2+
{L , L}
{L , O}
(1)
3+
where C (Ca ), L (La ), and O denote the two types of the metal cations and surface oxygens, respectively, and the indicators b and s denote whether at least one of the atoms is located in the bulk (b) or both belong to the surface layer (s). The quantities δXY are either 1, if the particles are of types X and Y, or 0 otherwise. Here the parameter wCL defines the energy of interactions between the nearest-neighbor pairs of Ca2+ and La3+ cations in bulk, wLL defines the interaction energy between the nearest and next-nearest neighbor pairs of surface La3+ cations unshielded by surface oxygens, and wLO defines the effective interaction energy of La3+ with the nearest surface O relative to that between Ca and surface O, which is defined as zero (wCO = 0). To calculate the model system properties, we used canonical Monte Carlo (MC) simulations detailed in the Methods section. Matching Structural Fingerprints by a Generative Model. To design a generative model with predictive abilities, it is critical to select the appropriate objective function that preserves the maximum amount of information about the target system’s fluctuations while limiting artifacts introduced by the model. As we have shown earlier on the design of atomistic force fields,38,48 such conditions are naturally fulfilled 721
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The resulting interaction parameters can be interpreted in terms of elastic and electrostatic effective interactions. A systematic study29 of contributions to dopant segregation in manganate perovskites predicted that the elastic contribution to the effective interactions can be expected to be small because the size of La3+ cations (1.172 Å) is only slightly larger than that of Ca2+ (1.14 Å),53 while the electrostatic effect on the surface segregation depended strongly on the considered model and ranged from positive to negative energy contributions. In agreement with this study,29 we identified unfavorable electrostatic interactions between the like cation pairs as a major, but not the only effect driving structure formation within the surface layer, and favorable elastic interactions, which prevail in the deeper layers and lead to the increased tendency to segregation. Besides these driving forces, we also analyzed the effect of surface oxygens on the metal atom distribution. As mentioned earlier, we observed that the apical O atoms preferentially attract La3+ particles, which seem to arrange into maze-like structures. The origin of this behavior can be associated with La aligning with higher probability at the opposite sides of the O atoms, as can be gleaned from the statistics of local configurations in Figures S4a−c. These histograms show that the probability of configuration 4, in which a pair of La atoms is located at the opposite sides of apical O atoms, exceeds about twice the probability of this configuration in a completely random structure, shown as red bars. In turn, this is reflected in the interaction parameter wLL = 6.28 kJ/mol of the optimized model, which quantifies the strength of the effective repulsive interaction between neighboring surface La atoms. This repulsion, leading to the formation of the linear and maze-like structures, can possibly be explained as resulting from the repulsion between increased electronic densities located between La and O atoms as seen in Figure 1e. Qualitatively, the linear arrangement of La atoms can be understood as resulting from the competition between the valence saturation of apical O atoms by formation of an optimal number of bonds with La atoms and the repulsion of the bonded La atoms. Similar structures have been observed in systems such as 2D binary alloys,54 where they result from frustration induced by the triangular geometry of their lattice. Even though cationic ordering at the A-site is rather unfavorable in general, it has been observed in select cases.55−57 Moreover, we emphasize that our observations are on the surface of a film and that only a tendency toward the formation of local structures exists, rather than a well-defined ordering in a crystallographic sense. As implied above, the model assumes that the effective interaction parameters are related to the observed structures through the Boltzmann distribution for a particular temperature. Even though we used the annealing (i.e., growth) temperature as a reference, as the sample was rapidly cooled at 150 C°/min after growth, the model predictions can be easily translated to any effective temperature by scaling the reduced interaction parameters. For instance, there could be residual motion of the Ca/La during the cool-down procedure. While we cannot be certain that our system reached the equilibrium, and there is an ongoing debate on the origin of surface segregation in manganites and the role of kinetic effects,58 experimental studies on the annealing of manganite thin films suggest significant atomic mobility and surface reconstruction above the film crystallization temperature, which occurs at around 1000 K.59 Also the ability of our simple model with
by a symmetric information measure known as statistical distance.35,49 In the present case of matching the histograms of target and model histograms of signature configurations, the distance measures the statistical distinguishability of the two sets of samples, and therefore its minimization results in models whose predictions are most easily mistaken for real measurements, that is, resulting in the type II error of statistical hypothesis testing. This inherent ability to separate signal from statistical noise is essential for identifying thermal and structural fluctuations in the presence of random errors caused by finite sampling, typical in microscopic image analysis. As follows from response theory,50 thermal fluctuations contain information about the response of the system to external perturbations (e.g., temperature, chemical potential), and thus their reproduction is essential for the model to be transferable and predictive. Statistical distance between two histograms with k bins is defined as k ji zy s = arccosjjjj∑ pi qi zzzz j z k i=1 {
(2)
where pi and qi are the relative frequencies of finding outcome i in the measurement of the experimental system P and model Q, respectively, with the total of k possible outcomes. Since all measurements are treated in a unified way as a collection of individual samples, rather than a specific physical quantity, they can be consistently integrated within this statistical framework. Here we match model predictions with the two complementary sources of information: the statistics of local configurations collected from image analysis and the concentration profile across layers from XPS measurement. The corresponding multitarget optimization loss function, equal to the sum of squares of partial statistical distances, is then51 2 2 2 S2 = cSOsSO + cSN sSN + cPRsPR
(3)
where the subscripts SO and SN denote surface configurations with and without oxygen, respectively, and PR denotes the concentration profile. The specific forms of the statistical distances s2 for different target data are given in the Supporting Information, Note 1. The coefficients cX, representing the relative statistical weights (uncertainty quantified by inverse variance) of the SO, SN, and PR data, were taken here as equal. Model Optimization. The optimal model parameters were found by minimizing the loss function in eq 3 using the perturbation technique.39,52 This histogram reweighting technique takes advantage of the thermodynamic perturbation relations to predict the changes of the model-generated histograms of local configurations and concentration profiles without needlessly repeating the actual simulations. The landscapes of the statistical distance (S2), whose minimum we seek, are shown as functions of the interaction parameters in Figure S5. The optimization yielded effective interactions in reduced units (w′ = w/kBT) wLC′ = 0.194, wLL′ = 0.739, and wLO′ = −0.890. Assuming that the system was equilibrated at the synthesis temperature of 1023 K, we can assign absolute values to the parameters as wLC = 1.65 kJ/mol, wLL = 6.28 kJ/ mol, and wLO = −7.57 kJ/mol. As can be seen in Figure 2a−c, the resulting model can reproduce quantitatively the target distributions of local surface configurations as well as the concentration profiles derived from experimental data. The sample configurations of the surface and bulk layers generated by the model are shown in Figure 2d,e, respectively. 722
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Figure 3. Model predictions: The thermodynamics and structural properties of LCMO thin film as a function of reduced inverse temperature and elemental composition. (a) Reduced Helmholtz free energy of elemental segregation, βF, and (b) excess entropy of elemental segregation, Sex/kB. (c) The mean number of atoms in finite Ca clusters, χ. (d) Probability p of Ca being a part of infinite cluster (spanning the periodic simulation box). White plus sign indicates our experimental conditions.
transitions within the probed range of conditions. Of course, given the positive energy of the La−Ca interactions represented by parameter wLC, the model will predict phase separation at lower temperatures. It has been shown experimentally that phase separation in bulk binary manganites can occur at higher temperatures,61 although here we are dealing with a thin film with two interfaces, with at least one of them stabilizing an even distribution of Ca and La atoms. The structural characteristics of the elemental segregation in terms of cluster formation for varying temperature and composition are summarized in Figure 3c,d, showing the mean size of finite clusters, χ, and the probability p of Ca particle belonging to an infinite cluster spanning the simulated system. Here we defined a Ca cluster as a maximal group of Ca atoms that are connected through the nearest-neighbor interactions and its size as the number of connected Ca atoms. We can see that at the experimental conditions (indicated by a white plus sign) Ca atoms are with 90% probability a part of a spanning cluster, with the rest mostly in 1- or 2-particle clusters. Such predictions for different conditions can be used for rational design of materials with desired level of elemental segregation, percolation, or other microstructural characteristics. The coarse-grained, effective character of the model interactions also means that the uncertainty of its predictions will grow for conditions further from the experimental temperature and composition. Since the higher-order compositional fluctuations, which provide information about the more distant parts of the phase diagram, are more difficult to distinguish from random noise resulting from the sampling uncertainty, the model predictions should be ideally complemented by additional measurements in an iterative experimental design cycle.
only three adjustable parameters to explain the available experimental data provides some support to the notion that the concentration fluctuations are driven by thermodynamic forces. Nonetheless we leave open the possibility that kinetic factors may also play a role in these systems, and efforts and modeling such dynamical fluctuations are ongoing. Predicting Subsurface Segregation and the Thermodynamic Response. The main value of the empirical generative model is its ability to generalize and make computationally efficient predictions of the system properties at a range of conditions. For instance, it allows us to reconstruct a phase diagram or identify conditions at which microstructures with desired properties form. The exploration of the property space of the LCMO film as a function of synthesis temperature, T, and Ca2+ fraction, x, was facilitated by special reweighting techniques based on the multistate Bennet’s acceptance ratio method.60 This approach allows us to perform a limited number of simulations (∼20) at different T and x and then extrapolate the properties to other conditions to achieve continuous phase space diagrams, as shown in Figure 3. Besides the Helmholtz free energy of the thin-film system, F, we calculated the corresponding excess entropy (per particle), Sex, defined as the difference between the full entropy and the entropy of a noninteracting system (ideal mixture) of the same Ca fraction x: S ex = S − Sid = S + xkB ln x + (1 − x)kB ln(1 − x)
(4)
The decrease of excess entropy in Figure 3b, for intermediate fractions x and lower temperatures (i.e., high reduced inverse temperature β*) indicates tendency for structuring, but the plot of the free energy in Figure 3a shows no signs of multiple minima corresponding to phase 723
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Figure 4. Variational autoencoder for phase diagram prediction. (a) Latent parameter as a function of composition, for different temperatures (colored). A minimum at the x = 0.5 concentration is found. A configuration sampled from the x = 0.375 and x = 0.50 data are shown in (b) and (c). (d and e) Plot of the surface and bulk x−y slices from these configurations (La, purple; Ca, yellow).
Unsupervised Exploration of the Phase Diagram. The 3D structures generated by the optimized model can be further processed by unsupervised ML methods to determine potentially interesting regions of the composition-temperature space. For example, even in the absence of macroscopic phase transitions, there may be anomalies in local ordering at certain compositions or temperatures, whose detection would normally require a large number of samples and macroscopic measurements. This can be done in two ways: One method relies on trying to form particular structural descriptors (as done above for the Ca clusters) and observing their changes with composition and temperature. An alternative method is to use a data-driven approach, where features of note can be “learned” via training on input data and is useful in situations where we are simply exploring the system and do not know the features that may be of interest. Of the several available methods for this task, here we tested the use of a variational autoencoder,31 which belongs to the class of neural networkbased algorithms that have been shown to be useful in data compression, image generation, and recently learning order parameters for phase transitions.62−64 The autoencoder consists of the “encoder” subsystem, which passes the input (here an atomic configuration) through a bottleneck and encodes it in a small number of latent parameters that contain compressed information about the input, and a “decoder” subsystem, which translates the latent parameters to output that should closely reproduce the input data.30,65 Details of the autoencoder network are available in the Methods section. We note here that this method is data driven, and the latent parameter learned need not have a physical meaning per se. During the training phase, batches of configurations are passed as input to the autoencoder, and the weights of the network are adjusted such that the input configurational data are reproduced by the network. In effect, the network learns to represent the configurational data as a small number of latent
parameters. The value of the latent parameter can be sampled from its probability distribution and passed through the decoder to produce new configurations. More importantly, the learned latent parameter(s) can be plotted as a function of temperature and composition, to observe any peculiar features. It is expected that as this parameter (in some manner) encodes information on the atomic configurations, then it should display anomalies when the configurations would change abruptly, as would be the case in, for example, phase separation. The results of the autoencoder output are shown in Figure 4(a), with the plot of the latent parameter as a function of concentration and effective temperature. Note that we used a single latent parameter. Although no clear trends can be seen along the temperature axis, the concentration dependence shows structure. There is a clear change in trends at the concentration value of x = 0.5 and an inflection around the experimental stoichiometry of x = 0.375. At this overall composition, the surface stoichiometry reaches the symmetric value of x ≈ 0.5, which can be associated with higher entropy. Interestingly, the latent variable appears to reproduce the qualitative trends in the excess entropy landscape in Figure 3b, which also exhibits a slight entropic increase for the experimental stoichiometry and lower temperatures. Again, we emphasize there does not need to be a direct relationship between the latent parameter and the physics of the system, but nonetheless, this match with the entropy is suggestive and is consistent with other reports.63,64 To examine this behavior more closely, we show the surface and bulk configurations for concentrations of x = 0.375 and x = 0.5 in Figure 4d,e. Visual comparison of the two surface structures in Figure 4d,e shows little change relative to bulk, indicating entropic stabilization of the x = 0.5 surface stoichiometry. We see that the variational autoencoder can be useful as an indicator of potential areas of 724
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PHOIBOS 150 hemispherical electron energy analyzer at room temperature in UHV at a base pressure
