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To be submitted to Applied Physics Letters Big Data in Reciprocal Space: Sliding Fast Fourier Transforms for Determining Periodicity Rama K. Vasudevan1,2,*, Alexei Belianinov1,2, Anthony G. Gianfrancesco3, Arthur P. Baddorf1,2, Alexander Tselev1,2, Sergei V. Kalinin1,2,3 and S. Jesse1,2 1 2
Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge TN 37831
Institute for Functional Imaging of Materials, Oak Ridge National Laboratory, Oak Ridge TN 37831 3
UT/ORNL Bredesen Center, University of Tennessee, Knoxville TN
Abstract Significant advances in atomically resolved imaging of crystals and surfaces have occurred in the last decade allowing unprecedented insight into local crystal structures and periodicity. Yet, the analysis of the long-range periodicity from the local imaging data, critical to correlation of functional properties and chemistry to the local crystallography, remains a challenge. Here, we introduce a Sliding Fast Fourier Transform (FFT) filter to analyze atomically resolved images of in-situ grown La5/8Ca3/8MnO3 films. We demonstrate the ability of sliding FFT algorithm to differentiate two sub-lattices, resulting from a mixed-terminated surface. Principal Component Analysis (PCA) and Independent Component Analysis (ICA) of the Sliding FFT dataset reveal the distinct changes in crystallography, step edges and boundaries between the multiple sub-lattices. The method is universal for images with any periodicity, and is especially amenable to atomically resolved probe and electron-microscopy data for rapid identification of the sub-lattices present.
*E-mail:
[email protected]
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Introduction The structure and morphology of surfaces and interfaces play a critical role in the properties of epitaxial heterostructures and materials in general. As a result of surface symmetry breaking, polarity discontinuities1 and cation segregation2 can lead to variation in properties ranging from metallic surface states3 in bulk insulators to complex flux-closure polarization patterns4 in ferroelectrics. For some materials classes, properties of the bulk are heavily dependent on the surface crystal structure (which can differ drastically from the bulk due to surface reconstruction), and are dictated by the interplay between the bulk properties and the chemical, charge and defect gradients at interfaces.5 It is therefore critical to understand surfaces and interfaces for deterministic materials design, with atomic-level imaging being a segue for correct identification of the structure and chemical nature of the surface. Although macroscopic scattering techniques can be used for both identification of surface species as well as material segregation and atomic ordering, these techniques typically average results over large macroscopic volumes of material and can miss the local ordering at the interface. As a result electron and scanning probe-based microscopy techniques are used to determine (via real-space imaging) the nature of the atomic ordering at the surface and across hetero-epitaxial interfaces. These imaging techniques provide atomically-resolved images allowing specific properties to be correlated with the imaged chemical nature and ordering (or lack there-of), with examples ranging from the nature of polarization in nominally ferroelectric6 and non-polar7 oxide films, to the segregation of vacancies in NiO8 that determine electroresistive behaviors9. However, while atomic imaging is widespread, quick and quantitative analysis of surface long-range periodicity and rapid identification of all the reconstructions in an image stack are lacking. Sliding FFT Algorithm Here, we employ a method based on Fast Fourier Transform (FFT) algorithm to rapidly analyze periodic images, with minimal user input, and show the utility of this approach on atomically resolved 3
images of La5/8Ca3/8MnO3 (LCMO) obtained in-situ by Scanning Tunneling Microscopy (STM). We note that use of the sliding windows is common in the computational sciences, e.g. in texture analysis where images are decomposed based on a set of exemplar textures10, 11. In our method, a window is rastered over the real space image with the FFT of the image subset within the window computed and stored. Subsequently, the window is shifted across the entire image until the entire original image has been scanned with all the FFTs fragments stored. Mathematically, a square window (size xa, ya) is defined with the upper left vertex initially located at the first (xi, yi) pixel location (see Fig. 1(a)) of a 2D image of width w and height h. The (2-D) Fast Fourier Transform is computed for the image portion within the window (with or without a boxcar filter), and the window is then advanced in (xs, ys) steps across the x-axis and y-axis, to a new location where the FFT is once again computed. The window is scanned for positions (xi,yi) with xi = 1,1+xs, …, 1+ Nx*xs, where Nx = (w-xa)/xs and similarly for yi = 1, 1+ ys, ...1+Ny*ys with Ny = (h-ya)/ys. For simplicity we enforce a step size xs (ys) that is a factor of both xa (ya) and w (h), i.e. the image width (height) and the window width (height). Note that the window is advanced across the x-axis as a sub-loop of steps of the window in the y-axis. As a result the FFT window (after all x and y steps) covers all areas of the image, with the final data matrix consisting of a stack of Nx*Ny Fast Fourier Transforms that contain all the information about the (spatially dependent) periodicity in the image. Note also that the window size, rather than the step size is the critical parameter since increasing the number of steps will not increase information content (but will only serve to interpolate pixels). The size of each individual 2D FFT can be modulated through the use of pixel interpolation, to smooth the FFT data, which increases the size of each FFT fragment by the interpolation factor (I). Finally a zoom factor (Z) can be employed, to eliminate edge effects in the FFT. Therefore the final size of the FFT images scales as I/Z. The final dataset is a 4-dimensional matrix of [Nx, Ny, kx, ky] with kx, ky being the dimensions of each individual FFT, i.e. kx = xa*I/Z and ky = ya*I/Z.
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Multiple Lattices on LCMO As a test case we applied the scanning FFT method on Scanning Tunneling Microscopy (STM) images of in-situ pulsed laser deposition grown LCMO films. More information about film growth and terminations will be published elsewhere12. For small thickness (
