Phases and Interfaces from Real Space Atomically Resolved Data

Aug 12, 2016 - Advances in electron and scanning probe microscopies have led to a wealth of atomically resolved structural and electronic data, often ...
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Letter pubs.acs.org/NanoLett

Phases and Interfaces from Real Space Atomically Resolved Data: Physics-Based Deep Data Image Analysis Rama K. Vasudevan, Maxim Ziatdinov, Stephen Jesse, and Sergei V. Kalinin* Institute for Functional Imaging of Materials and Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge Tennessee 37831, United States S Supporting Information *

ABSTRACT: Advances in electron and scanning probe microscopies have led to a wealth of atomically resolved structural and electronic data, often with ∼1−10 pm precision. However, knowledge generation from such data requires the development of a physics-based robust framework to link the observed structures to macroscopic chemical and physical descriptors, including single phase regions, order parameter fields, interfaces, and structural and topological defects. Here, we develop an approach based on a synergy of sliding window Fourier transform to capture the local analog of traditional structure factors combined with blind linear unmixing of the resultant 4D data set. This deep data analysis is ideally matched to the underlying physics of the problem and allows reconstruction of the a priori unknown structure factors of individual components and their spatial localization. We demonstrate the principles of this approach using a synthetic data set and further apply it for extracting chemical and physically relevant information from electron and scanning tunneling microscopy data. This method promises to dramatically speed up crystallographic analysis in atomically resolved data, paving the road toward automatic local structure−property determinations in crystalline and quasi-ordered systems, as well as systems with competing structural and electronic order parameters. KEYWORDS: Unmixing, crystallography, atomic scale imaging, Fourier transform, scanning tunneling microscopy, scanning transmission electron microscopy

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remain inaccessible. Further complicating efforts at appropriate structure determination are the presence of multiple coexisting phases, as can be the case for numerous catalytic materials, or when the interface between distinct phases is of special importance to the functional property, as is the case for, e.g., the triple phase boundary in fuel cells.12,13 In parallel, difficulties can arise in disordered materials consisting of numerous microdomains which are small, leading to situations where averaging over macroscopic volumes of material can produce difficult to interpret spectra in X-ray or neutron scattering that can mask the true underlying physics.14,15 Examples such as inhomogeneous and nanoscale phase separated systems, structural and topological defects, interfaces, and polycrystalline materials necessitate the development of a language and knowledge base to describe local crystallographic properties.11,16 While in some cases scattering approaches rooted in symmetry-based physical models can be extended to systems such as grain boundaries and domain walls,17−19 in general such descriptions lead to rapid growth in the dimensionality of the parameter space (e.g., five parameters

he link between crystal structure and material functionality is a fundamental cornerstone of modern science, with examples ranging from displacive atomic motions in austenite− martensite transformations triggered by applied stress in shapememory alloys,1 to changes in bond length in carbon compounds altering the molecule’s reactivity,2 and bond angles in perovskite manganites intrinsically linked to magnetic and transport properties.3,4 The research field of crystallography has been vital for researchers in chemistry, physics, and materials science since its birth in the early 1900s and has been used to map the crystal structure of a vast array of compounds and molecules5 with picometer precision. The accumulation of this data has led to a rich knowledge base that links the crystal structure to the measured properties, including elastic,6 electric,7 thermal,8 magnetic,9 etc., and resulting in quantum leaps in materials technology, biology, and physics that form the bedrock of the modern age. Although even the most complex of crystal structures, such as large biomolecular crystals,10 are amenable to this classical approach, the determination of point groups is insufficient to completely describe even the simplest of structures that is somewhat disordered, ranging from partial site occupancy to inhomogeneous ground states.11 While statistically averaged occupancies can be obtained from scattering data, local states © 2016 American Chemical Society

Received: May 26, 2016 Revised: July 22, 2016 Published: August 12, 2016 5574

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Figure 1. Schematic of analysis technique. Extraction of phases in an atomically resolved image begins with processing, which involves the sliding fast Fourier transform to generate a stack of 2D FFT images. Analysis proceeds by reducing the dimensionality through projection (e.g., by PCA), and then utilizing an endmember extraction algorithm, such as N-FINDR, which can determine the pure spectra present, along with the relative spatial abundance.

some existing real-space methods, as it does not require fitting to unique image features or image elements,16 and will work so long as periodicity (not necessarily atomic-scale) is present. Such a technique would be a first step toward the establishment of a structure−property database for local crystallography, analogous to the macroscopic crystal structure databases that previous decades have endowed the community with, and would be a critical tool for future material developments.27,28 Here, we develop a robust, physics-based method for determination of a priori unknown local abundances of phases from atomically resolved imaging data. In this approach, the structure of the material at each location is identified by a classical Fourier transform within a local window. Sliding the window along the image plane generates a stack of 2D FFT images, yielding a 4D structural image. This stack is then analyzed by means of an endmember extraction algorithm (NFINDR) to determine the phases present in the image (i.e., the “pure” spectra, termed the “endmembers”), as well as the spatial abundance of the phases present. This technique is verified using a model synthetic data set and further extended to analyze chemically and physically heterogeneous systems, directly mapping phases and order parameter fields. The effects of systematically varying the parameters of the algorithm, including the window size and number of endmembers is determined (see Supporting Information), with results indicating that an optimum value for the endmembers is reached by simple inspection of similarity of endmembers, or use of calculated error maps, while the window size optimum is set by the characteristic size and relative abundance of the domains (or phases) present and can be determined from scaledependent dispersion of the stack. This analysis thus offers a robust local, real-space technique somewhat analogous to structure factor determination from diffraction experiments. Moreover, we suggest these algorithms will ultimately form part of the input for automatic detection of structure−property relations for entries in materials genome-type libraries.

required to describe a general grain boundary20), and the relevant descriptors become experimentally inaccessible from scattering studies. However, the situation changed dramatically with the proliferation of real-space high resolution imaging studies. The explosion of scanning transmission electron microscopy (STEM) as well as scanning tunneling microscopes (STM) has produced an enormous quantity of atomically resolved images of technologically relevant materials, and throughput is becoming ever larger in the age of big data science.21,22 Indeed, approximately picometer precision in atomically resolved STEM images is now available, and tens of gigabytes of image data can be sourced from a single microscope in one sitting. Since the key chemical and physical properties of crystals are inherently interlinked with the positions (or small distortions) of the atoms comprising the unit cell, high-veracity imaging data of this type present a tremendous opportunity for data mining to extract information relevant to mapping the structure to chemical or physical properties. However, methods to analyze the generated atomicscale images to determine crystal structures present, and their relative spatial abundance, are limited and yet are increasingly necessary as the volumes of data expand beyond the researcher’s ability to comprehend and analyze them. Analysis to date has focused on techniques such as geometric phase analysis,23 where the fast Fourier transform (FFT) of the entire image is computed and subsequently filtered and processed to reveal displacement and strain maps, and variational mode decomposition,24 which has very recently been developed to deduce harmonic patterns in images. Furthermore, recent work by Eggeman et al.25 showed the utility of algorithms such as non-negative matrix factorization in unmixing spectral images, in their case from precession electron diffraction, to determine different structural phases in a Ni-based superalloy. However, the sliding window approach, which is common in texture analysis,26 has not been employed for tackling these problems. Such an approach has the advantage of greater generality than 5575

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determine which pixels in the image constitute endmembers. Specifically, the algorithm N-FINDR, developed by Winter,30,31 is utilized which allows solving the mixing problem:

The process of determining the phases present, along with their spatial abundance, is split into two steps, as indicated in Figure 1. At the first step, the image is analyzed via the sliding FFT method, which has been described in an earlier publication.29 Essentially, it involves forming a window of size (wx, wy) that is stepped across the image I (image size Ix, Iy), in a series of steps xs and ys such that the entire image is scanned. At each step, the FFT of the image that lies within the bounds of the window is computed, resulting in a stack of 2D FFT images at the conclusion of the stepping process. Filtering, interpolation, and zooming during the FFT process can be performed, and in this manuscript, all of the studies conducted involve the use of a Hanning window to minimize edge effects, as well as a 2 zoom combined with a 2× interpolation function to increase pixel density during the FFT capture step. This analysis step has a link to phase determination from Xray diffraction of crystals. Namely, in a scattering experiment, the crystal phases are identified by their spectra (in our case, 2D FFT spectra) which (for diffraction) can be calculated via the structure factor Fs:

N

p(x , y ) =

k=1

∑ fi e 2πi(hu + kv + lw ) i

i=1

j

N

∑ c k (x , y ) = 1 k=1

i

(1)

N

∑ ciFi i=1

(4)

which of course posits that the phases are fractions that sum to one, and furthermore, we apply the non-negative constraint, ideally matching the physics of the system. It is assumed that pure endmembers are present in the spectral data (which physically means that pure FFTs of the different lattice types were captured), thus necessitating that the window size chosen is smaller than the characteristic size of the domains present, but larger than the unit cell size. In the case that this condition is not met, the algorithm will still compute but will instead assign the pixel that is closest to being “pure” as the endmember. Here we draw attention to the fact that the endmembers need not be known, unlike the diffraction case which requires positing the atomic positions of the components before regression analysis on the predicted and measured (squared) structure factor is performed.32 Furthermore, even when the pure pixels are absent, more complex methods such as hierarchical Bayesian models can be used to determine the individual elements;33 hence this condition, while making the computational cost much lower, is nonessential. The spectral end-members (i.e., image structure factors of pure phases) are obtained via the N-FINDR algorithm. It should be noted that the image structure factor data is in fact the modulus of the complex FFT data. It begins by projecting the data into a smaller subspace, generally through the use of principal component analysis (PCA). The dimension of the subspace is (N − 1), where N is the number of endmembers to be determined. The volume of the simplex formed by pixels randomly chosen as endmembers is then calculated. Iteratively, the algorithm searches for the set of pixels such that the largest possible volume of the simplex results, as these would correspond to the pixels that comprise the purest endmember spectra. Inspection of the endmembers provides the information on the structure of pure phases present in the image, and the abundance maps produced provide quantitative information on the (real-space) spatial distribution of the phases themselves. Here, the “structure” that is being determined is essentially the reciprocal lattice of the surface of the constituent phases. If the surface is not reconstructed, then one can easily derive the Bravais lattice primitives in two-dimensions for the particular phases, but determination of the crystallographic space group is considerably more challenging. Full extension of the unmixing to 3D (for instance, see refs 34 and 35), as well as determination of the space group, are technically possible, but the latter would require substantial efforts at developing the

where hkl are the Miller indices of the diffracting planes, uvw describes the positions of the atoms of the primitive unit cell, N is the number of atoms in the unit cell, and the total structure factor F0 describing the scattering is given as F0 =

(3)

where p = p(x, y) is the spectra at pixel (x,y), ek is the endmember and c = ck(x, y) are the position-dependent weighting coefficients, and ε is a Gaussian error, assumed to be small and N is the number of endmembers (user-defined). Note the similarity of eq 3 and eq 2, thus showing the analogue to a global diffraction experiment. Furthermore, because the weights are considered to be percentages, the following must be satisfied:

N

Fs =

∑ ekck(x , y) + ε

(2)

where ci is the fraction of phase with structure factor Fi, and the intensity of the spectra I ∝ |F0|2. Given the experimentally acquired diffraction intensity I, the fitting then proceeds via adjustment of the atomic positions (uvw) and components ci in the model, thus providing information on the phases and their abundances. For a multiple phase mixture with sufficiently large single-phase regions, the structure factor is a linear superposition of the individual constitutive elements. It should be noted that in the case of substantial dynamical effects, and e.g. contributions from multiple phases with different microstructures, the linear superposition is likely to be inaccurate, precluding further unmixing analysis posited here. Comparatively, in the local analysis the local 2D FFT is acquired, which is analogous to the structure factor, although as the source may not be from scattering, cannot be regarded as equivalent. We therefore term the local FFT spectra as an image structure factor, to distinguish this from the more well-known scattering structure factor. We also note here that the image structure factor is dependent on the input data, i.e., in cases where the input images do not coincide with the actual lattice structure (due to electronic phases in STM, or due to dynamical scattering effects in STEM), the analogy with the diffraction data is no longer valid. In the second stage, analysis of the acquired stack of FFT images is performed using tools that were originally designed for analysis of hyperspectral data in order to determine the “pure” spectra (or endmembers) present, i.e., to unmix the data into the constitutive phases, assuming the same linear mixing model of the image structure factors. Computationally, the image structure factors are assumed to be a linear superposition of the endmembers, and the task of the algorithm is to 5576

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Figure 2. Ideal test case (computer generated). (a) A test case, where two different lattice types are in the same image, is shown. The left half of the image contains a diamond lattice type, while the right half displays a square lattice type. The window size chosen for the sliding FFT algorithm is 64px and is drawn as a white dashed box for guide to the eye. Inset, a global FFT of the image is shown, where mixing of the two (spatially) distinct phases is apparent. (b−e) N-FINDR results (for nc = 2) after sliding FFT was carried out on the test case. (b,c) The two endmembers and (d,e) spatial abundances. The two lattice types are clearly visible in the FFTs, and the unmixing in this case is very precise.

Figure 3. Real STEM image of a catalyst. (a) STEM image, with the window from the sliding FFT overlaid in white. The scale bar is 10 nm. Inset, global FFT. Results of endmember extraction with N-FINDR (N = 3) after sliding FFT, for the atomically resolved image in panel a are shown in b− g. (b−d) The three endmembers and (e−g) their spatial abundances are shown. Clearly both phases are isolated by the algorithm. The width of the window is 500px, and the window step size is 100px. Image in panel a is reprinted with permission from He et al.36 Copyright 2015 American Chemical Society.

appropriate image classification schemes and are left for future study. To ascertain the applicability and robustness of this method, we created a test image consisting of two different lattice types, with the left half of the image containing a (√2 × √2)R45° pattern and the right half consisting of a (1 × 1) square lattice. The test image, along with the outline of the window used in the sliding FFT step, is shown in Figure 2a. The global 2D FFT is shown as an inset in Figure 2a, and the reciprocal space (diffraction) spots that define both lattice types are clearly visible in the inset FFT image. After the FFT images were captured, the data were unmixed using the N-FINDR approach (with N = 2), and the results are shown in Figure 2b−e. The

weighting coefficients for the endmember spectra are shown in the abundance maps in Figure 2d,e and clearly demarcate the two regions, showing ideal separation. This result confirms the applicability of the sliding FFT method in conjunction with endmember extraction via the N-FINDR approach in order to determine the types and spatial abundances of phases present in an atomically resolved image. The determination of the optimum number of endmembers as well as the optimum window size is discussed in the Supporting Information. Having verified that our algorithm is capable of working on an ideal test case, we moved to a real STEM image of a Mo− V−Te−Nb oxide catalyst containing two distinct phases. More details and analysis of this image have been published 5577

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Figure 4. Simulated image and unmixing for dimer-hexagonal (super)lattice. (a) Test image with a well-defined rectangular domain of a dimer lattice in the right-side bottom of the image, and separate dimer features embedded locally into the hexagonal lattice in the rest of the image. (b,c) Two endmembers and (d,e) spatial abundances (image size 512 px × 512 px, window size 64 px, and step size 16 px). (f,g) Real-space image of ideal hexagonal (f) and dimer (g) lattices and their corresponding FFTs shown in the insets. Orange arrows in b,c and f,g denote the two spots in the inner hexagon that become strongly suppressed for a dimer lattice.

elsewhere.36 The STEM image is shown in Figure 3a, and the global FFT is shown inset. The size of the window used in the sliding FFT method is outlined with a white dotted line. For the N-FINDR analysis, we begin by postulating three endmembers and observing the results, which are shown in Figure 3b−g. The algorithm clearly identifies at least two distinct phases, and a third phase is identified as the blank space in the image which is characterized mainly by high noise, i.e., a lack of periodicity leading to a large central spot in the FFT spectra with little other distinguishable features. Results for this image with variations of the window size and number of endmembers are provided in the Supporting Information and show that an optimum exists for a particular window size, which can be characterized mathematically. In addition, inspection of the endmembers in Figure 3 reveals the types of phases that are present in the Mo−V−Te−Nb oxide system. While the matrix (“M2” phase) is hexagonal, the inclusion (“M1” phase) displays an FFT that is more complicated. In fact, this M1 phase has a large unit cell consisting of both heptagonal and hexagonal channels, and it was found that the M1 phase was associated with the catalytic ability of this oxide.37 It also appears that the M1 phase can be seen as a region of low (∼10%) intensity within the first abundance map (Figure 3e). This effect may be due to an artifact as a result of the unit cell size being substantially larger than the M2 phase, leading to greater intensity of the central FFT peak (Figure 3b) for a predetermined window size. These results reiterate the need to be cognizant of the dynamical effects as well as artifacts that can impact the unmixing. Nevertheless, if these effects are small, they are not expected to substantially impact the final segmentation. Thus, in situations where one phase is associated with the functional property,

methods such as this analysis will allow for automatic detection and mapping of the key structural elements, speeding up analysis and providing quantitative metrics for comparing, e.g., different preparation methods. We next discuss the application of our method to STM images of layered materials with a complex electronic structure. In the initial examples, the hypothesis was that the atomically resolved image consisted of exclusive phases which could not coexist in the same region. However, unlike STEM data, the protrusions in the STM image reflect the spatial distribution of electronic densities near the Fermi level and may not necessarily have a simple relationship to the underlying atomic lattice, and there the hypothesis of exclusive phases is no longer valid, as electronic phases can often coexist. Furthermore, the electronic (super)lattices frequently have a complex fine structure associated with an electron scattering at the multiple impurities (e.g., see refs 38−41) or with locally varying strain fields.42 We focus below on the fine structure of the electronic superlattice in a correlated Mott insulator α-RuCl3 formed by modulations of a charge density in a form of dimer and hexagonal patterns superimposed on the regular atomic lattice. The detailed discussion on the physical nature of the electronic density patterns in this material will be published elsewhere. We start with a construction of a test image (Figure 4a), in which we place a well-defined rectangular domain of a dimer lattice in the bottom right corner of the image and then implant the individual dimer structures locally into the hexagonal lattice that constitutes the remaining part of the test image. The resultant endmembers, corresponding abundance maps, and the separate FFTs of each superlattice are shown in Figure 4b,c, d,e, and f,g, respectively. The clear reciprocal space signature of the dimer lattice formation is a strong suppression of two spots in 5578

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Figure 5. Experimental results of unmixing applied to STM image of α-RuCl3. (a) Experimental STM constant current image obtained on α-RuCl3 surface; tunneling conditions: Utip = −300 mV, Iset = 60 pA. Coexisting dimer and hexagonal superlattices are magnified in the inset. (b,c) Two endmembers and (d,e) spatial abundances (image size 512 px × 512 px, window size 64 px, and step size 16 px). Orange arrows in b,c have the same meaning as in Figure 4.

The uniqueness of our approach lies in the matching of the unmixing problem to the physics of the system, which translates to applying the appropriate constraints on the endmembers. It should be noted that, although factorization algorithms can be utilized for similar unmixing problems, the endmember extraction algorithms are ideally suited for the problem, as they assume (generally) that pure pixels exist within the data set, and the spectral weights are percentages, allowing the sum to one constraint to be applied. The process of automatic phase identification and assignment will become more important with the advent of genomic libraries for materials,21,27 to determine the structure−property relations at the atomic scale. The tool elucidated here can be combined with identification with lattice types, albeit taking heed of possible arbitrary rotation of the lattice/s. These approaches can also be applied to quasi-ordered45,46 systems or multicomponent systems where some phases are amorphous or only quasi-crystalline. Further, they show potential to help solving fundamental problems in condensed matter physics, e.g., by analyzing quasiparticle scattering interference and modulations in electronic density of states in high-temperature superconductors and topological insulators.47 For example, presence of magnetic and nonmagnetic impurities on the surface of a topological insulator could cause two distinct electron scattering mechanisms, which would be visible as a local enhancement and suppression of specific wavevectors in FFT space. Given that the real space distribution of strength and types of quasi-particle scattering cannot be revealed by any other method, the method we present is ideal to analyze such data. Finally, combined with local crystallography analysis by individual atom identification and classification,16 these methods can constitute a framework for the language of correlated disorder, as recently outlined by Goodwin and Keen.11 Experimental Section. The sliding FFT method was implemented in Python 2.7 with PySptools for the NFINDR algorithm (http://pysptools.sourceforge.net/). The STM im-

the inner hexagon (denoted by arrows) in the FFT spectra. Inspection of abundance maps in Figure 4d,e shows that not only do the abundance maps reveal unambiguously a presence of the rectangular domain of the dimer lattice, but the method is also sensitive enough to distinguish the diluted concentration of dimer features embedded in the hexagonal lattice in the rest of the image (see Figure 4e). This observation is of particular importance for mapping a nanometer-scale fine structure of the complex electron density lattices. We now turn to the experimental STM images of the empty electronic states on α-RuCl3 surface that contain both dimer and hexagonal charge ordered patterns (Figure 5a). In a good agreement with the analysis of the test image above, we were consistently able to find the endmember in which two spots of the inner hexagon are strongly suppressed reflecting a formation of a dimer lattice in addition to the hexagonal electron densities lattice (Figure 5b,c). The associated abundance maps are shown in Figure 5d,e. Overall, a successful separation of atomic and electronic lattices in STEM and STM data sets suggests a universality of the method for microscopic lattice images with a subnanometer resolution. The approach outlined in this manuscript shows a robust method to identify the presence and spatial abundance of crystallographic and structural/electronic phases automatically with minimum user input. The choice of number of endmembers can also be automated to be large (e.g., 20), with the algorithm cutting off once either the error maps become far from ideal, the endmembers become mixed or duplicated, or both. More sophisticated approaches can also be utilized, such as the Harsanyi−Farrand−Chang (HFC) method to estimate the number of endmembers before beginning the algorithm, extending the automation of the process.43,44 Furthermore, optimization of the parameters within the sliding FFT can be selected based on input of characteristic domain size or automated through local FFT analysis at random locations, or other more sophisticated techniques (extended discussion in Supporting Information). 5579

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(16) Belianinov, A.; He, Q.; Kravchenko, M.; Jesse, S.; Borisevich, A.; Kalinin, S. V. Nat. Commun. 2015, 6, 7801. (17) Sutton, A. P.; Balluffi, R. W. Clarendon Press, 1995. (18) Tagantsev, A. K.; Cross, L. E.; Fousek, J. Domains in ferroic crystals and thin films; Springer, 2010. (19) Wadhawan, V. Introduction to ferroic materials; CRC Press, 2000. (20) Lejcek, P. Grain boundary segregation in metals; Springer Science & Business Media, 2010; Vol. 136. (21) Kalinin, S. V.; Sumpter, B. G.; Archibald, R. K. Nat. Mater. 2015, 14 (10), 973−980. (22) Managing your microscopy big image data: Challenges, strategies, solutions. In Technology Webinars; Science/AAAS Custom Publishing Office, 2015. (23) Hÿtch, M.; Snoeck, E.; Kilaas, R. Ultramicroscopy 1998, 74 (3), 131−146. (24) Dragomiretskiy, K.; Zosso, D. In Two-dimensional variational mode decomposition, Energy Minimization Methods in Computer Vision and Pattern Recognition; Springer, 2015; pp 197−208. (25) Eggeman, A. S.; Krakow, R.; Midgley, P. A. Nat. Commun. 2015, 6, 7267. (26) Tuceryan, M.; Jain, A. K. Handbook of pattern recognition and computer vision 1993, 2, 207−248. (27) Jain, A.; Ong, S. P.; Hautier, G.; Chen, W.; Richards, W. D.; Dacek, S.; Cholia, S.; Gunter, D.; Skinner, D.; Ceder, G. APL Mater. 2013, 1 (1), 011002. (28) Sumpter, B. G.; Vasudevan, R. K.; Potok, T.; Kalinin, S. V. Npj Computational Materials 2015, 1, 15008. (29) Vasudevan, R. K.; Belianinov, A.; Gianfrancesco, A. G.; Baddorf, A. P.; Tselev, A.; Kalinin, S. V.; Jesse, S. Appl. Phys. Lett. 2015, 106 (9), 091601. (30) Winter, M. E. In N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data; SPIE’s International Symposium on Optical Science, Engineering, and Instrumentation; International Society for Optics and Photonics, 1999; pp 266−275. (31) Winter, M. E. In A proof of the N-FINDR algorithm for the automated detection of endmembers in a hyperspectral image; Defense and Security; International Society for Optics and Photonics, 2004; pp 31−41. (32) O’Neill, H. S. C.; Dollase, W. Phys. Chem. Miner. 1994, 20 (8), 541−555. (33) Dobigeon, N.; Moussaoui, S.; Coulon, M.; Tourneret, J.-Y.; Hero, A. O. Signal Processing, IEEE Transactions on 2009, 57 (11), 4355−4368. (34) Nicoletti, O.; de La Peña, F.; Leary, R. K.; Holland, D. J.; Ducati, C.; Midgley, P. A. Nature 2013, 502 (7469), 80−84. (35) Eggeman, A. S.; Krakow, R.; Midgley, P. A. Nat. Commun. 2015, 6, 7267. (36) He, Q.; Woo, J.; Belianinov, A.; Guliants, V. V.; Borisevich, A. Y. ACS Nano 2015, 9 (4), 3470−3478. (37) Ushikubo, T. Catal. Today 2000, 57 (3), 331−338. (38) Ziatdinov, M.; Fujii, S.; Kusakabe, K.; Kiguchi, M.; Mori, T.; Enoki, T. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89 (15), 155405. (39) Rutter, G.; Crain, J.; Guisinger, N.; Li, T.; First, P.; Stroscio, J. Science 2007, 317 (5835), 219−222. (40) Fujii, S.; Enoki, T. ACS Nano 2013, 7 (12), 11190−11199. (41) Fujii, S.; Ziatdinov, M.; Ohtsuka, M.; Kusakabe, K.; Kiguchi, M.; Enoki, T. Faraday Discuss. 2014, 173, 173. (42) Soumyanarayanan, A.; Yee, M. M.; He, Y.; van Wezel, J.; Rahn, D. J.; Rossnagel, K.; Hudson, E.; Norman, M. R.; Hoffman, J. E. Proc. Natl. Acad. Sci. U. S. A. 2013, 110 (5), 1623−1627. (43) Plaza, A.; Chang, C.-I. In An improved N-FINDR algorithm in implementation, Defense and Security; International Society for Optics and Photonics, 2005; pp 298−306. (44) Chang, C.-I.; Du, Q. Geoscience and Remote Sensing, IEEE Transactions on 2004, 42 (3), 608−619. (45) Bodnarchuk, M. I.; Shevchenko, E. V.; Talapin, D. V. J. Am. Chem. Soc. 2011, 133 (51), 20837−20849.

ages were captured at room temperature on UHV-cleaved RuCl3 single crystals using mechanically cut Pt/Ir tips on an Omicron VT STM/AFM. Details about the STEM image analyzed can be found in another publication.36

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b02130. Additional results on the performance of the algorithm, optimization of parameters, as well as extended discussion (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank A. Borisevich (ORNL) and Q. He (ORNL) for permission to use the STEM image published previously for this manuscript.36 This research was sponsored by the Division of Materials Sciences and Engineering, BES, US DOE (RKV, MZ, SVK). Research was conducted at the Center for Nanophase Materials Sciences, which also provided support (S.J.) and which is a US DOE Office of Science User Facility.

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REFERENCES

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DOI: 10.1021/acs.nanolett.6b02130 Nano Lett. 2016, 16, 5574−5581