Quantitative analysis of compatible microstructure by electron backscatter diffraction.

We propose a scheme for assigning the martensite variant using electron backscatter diffraction in a martensite material that undergoes a solid-solid phase transformation. Based on the solutions of the crystallographic equations of martensite, we provide an algorithm to assign martensite variants to a particular microscopic region, and to check the elastic compatibility of the microstructure corresponding to low hysteresis and high reversibility in shape memory alloys. This article is part of the theme issue 'Topics in mathematical design of complex materials'.

Reflector Selection for the Indexing of Electron Backscatter Diffraction Patterns.

We propose a new methodology for ranking the reflectors used in traditional Hough-based indexing of electron backscatter diffraction (EBSD) patterns. Instead of kinematic X-ray or electron structure factors (Fhkl) currently utilized, we propose the integrated Kikuchi band intensity parameter (ßhkl) based on integrated dynamical electron backscatter intensities. The proposed parameter is compared with the traditional kinematical intensity, , as well as the average Hough transform peak intensity, and used to index EBSD patterns for a number of different material systems of varying unit cell complexities including nickel, silicon, rutile, and forsterite. For elemental structures, ßhkl closely follows the kinematical ranking. However, significant ranking differences arise for more complex unit cells, with the ßhkl parameter showing a better correlation with the integrated Hough intensities. Finally, Hough-based indexing of a simulated forsterite data set showed an appreciable improvement in the median confidence index (0.15 to 0.35) when ßhkl is used instead of for ranking the reflectors.

Lorentz Transmission Electron Microscopy Image Simulations of Experimental Nano-Chessboard Observations in Co-Pt Alloys.

The magnetization configuration of a novel nano-chessboard structure consisting of L10 and L12 phases in a Co40Pt60 alloy is investigated using Lorentz transmission electron microscopy (LTEM) and micro-magnetic simulations. We show high-resolution LTEM images of nano-size magnetic features acquired through spherical aberration correction in Lorentz Fresnel mode. Phase reconstructions and LTEM image simulations are carried out to fully understand the magnetic microstructure. The experimental Fresnel images of the nano-chessboard structure show zig-zag shaped magnetic domain walls at the inter-phase boundaries between L10 and L12 phases. A circular magnetization distribution with vortex and anti-vortex type arrangement is evident in the phase reconstructed magnetic induction maps as well as simulated maps. The magnetic contrast in experimental LTEM images is interpreted with the help of magnetic induction maps simulated for various relative electron beam-sample orientations inside the TEM.

Extracting Grain Orientations from EBSD Patterns of Polycrystalline Materials Using Convolutional Neural Networks.

We present a deep learning approach to the indexing of electron backscatter diffraction (EBSD) patterns. We design and implement a deep convolutional neural network architecture to predict crystal orientation from the EBSD patterns. We design a differentiable approximation to the disorientation function between the predicted crystal orientation and the ground truth; the deep learning model optimizes for the mean disorientation error between the predicted crystal orientation and the ground truth using stochastic gradient descent. The deep learning model is trained using 374,852 EBSD patterns of polycrystalline nickel from simulation and evaluated using 1,000 experimental EBSD patterns of polycrystalline nickel. The deep learning model results in a mean disorientation error of 0.548° compared to 0.652° using dictionary based indexing.

Dictionary Indexing of Electron Channeling Patterns.

The dictionary-based approach to the indexing of diffraction patterns is applied to electron channeling patterns (ECPs). The main ingredients of the dictionary method are introduced, including the generalized forward projector (GFP), the relevant detector model, and a scheme to uniformly sample orientation space using the "cubochoric" representation. The GFP is used to compute an ECP "master" pattern. Derivative free optimization algorithms, including the Nelder-Mead simplex and the bound optimization by quadratic approximation are used to determine the correct detector parameters and to refine the orientation obtained from the dictionary approach. The indexing method is applied to poly-silicon and shows excellent agreement with the calibrated values. Finally, it is shown that the method results in a mean disorientation error of 1.0° with 0.5° SD for a range of detector parameters.

Reconstruction of Laser-Induced Surface Topography from Electron Backscatter Diffraction Patterns.

We demonstrate that the surface topography of a sample can be reconstructed from electron backscatter diffraction (EBSD) patterns collected with a commercial EBSD system. This technique combines the location of the maximum background intensity with a correction from Monte Carlo simulations to determine the local surface normals at each point in an EBSD scan. A surface height map is then reconstructed from the local surface normals. In this study, a Ni sample was machined with a femtosecond laser, which causes the formation of a laser-induced periodic surface structure (LIPSS). The topography of the LIPSS was analyzed using atomic force microscopy (AFM) and reconstructions from EBSD patterns collected at 5 and 20 kV. The LIPSS consisted of a combination of low frequency waviness due to curtaining and high frequency ridges. The morphology of the reconstructed low frequency waviness and high frequency ridges matched the AFM data. The reconstruction technique does not require any modification to existing EBSD systems and so can be particularly useful for measuring topography and its evolution during in situ experiments.

Influence of Noise-Generating Factors on Cross-Correlation Electron Backscatter Diffraction (EBSD) Measurement of Geometrically Necessary Dislocations (GNDs).

Studies of dislocation density evolution are fundamental to improved understanding in various areas of deformation mechanics. Recent advances in cross-correlation techniques, applied to electron backscatter diffraction (EBSD) data have particularly shed light on geometrically necessary dislocation (GND) behavior. However, the framework is relatively computationally expensive-patterns are typically saved from the EBSD scan and analyzed offline. A better understanding of the impact of EBSD pattern degradation, such as binning, compression, and various forms of noise, is vital to enable optimization of rapid and low-cost GND analysis. This paper tackles the problem by setting up a set of simulated patterns that mimic real patterns corresponding to a known GND field. The patterns are subsequently degraded in terms of resolution and noise, and the GND densities calculated from the degraded patterns using cross-correlation ESBD are compared with the known values. Some confirmation of validity of the computational degradation of patterns by considering real pattern degradation is also undertaken. The results demonstrate that the EBSD technique is not particularly sensitive to lower levels of binning and image compression, but the precision is sensitive to Poisson-type noise. Some insight is also gained concerning effects of mixed patterns at a grain boundary on measured GND content.

Nanoscale Skyrmions in a Nonchiral Metallic Multiferroic: Ni2MnGa.

Magnetic skyrmions belong to a set of topologically nontrivial spin textures at the nanoscale that have received increased attention due to their emergent behavior and novel potential spintronic applications. Discovering materials systems that can host skyrmions at room temperature in the absence of external magnetic field is of crucial importance not only from a fundamental aspect, but also from a technological point of view. So far, the observations of skyrmions in bulk metallic ferromagnets have been limited to low temperatures and to materials that exhibit strong chiral interactions. Here we show the formation of nanoscale skyrmions in a nonchiral multiferroic material, which is ferromagnetic and ferroelastic, Ni2MnGa at room temperature without the presence of external magnetic fields. By using Lorentz transmission electron microscopy in combination with micromagnetic simulations, we elucidate their formation, behavior, and stability under applied magnetic fields at room temperature. The formation of skyrmions in a multiferroic material with no broken inversion symmetry presents new exciting opportunities for the exploration of the fundamental physics of topologically nontrivial spin textures.

Performance of Dynamically Simulated Reference Patterns for Cross-Correlation Electron Backscatter Diffraction.

High-resolution (or "cross-correlation") electron backscatter diffraction analysis (HR-EBSD) utilizes cross-correlation techniques to determine relative orientation and distortion of an experimental electron backscatter diffraction pattern with respect to a reference pattern. The integrity of absolute strain and tetragonality measurements of a standard Si/SiGe material have previously been analyzed using reference patterns produced by kinematical simulation. Although the results were promising, the noise levels were significantly higher for kinematically produced patterns, compared with real patterns taken from the Si region of the sample. This paper applies HR-EBSD techniques to analyze lattice distortion in an Si/SiGe sample, using recently developed dynamically simulated patterns. The results are compared with those from experimental and kinematically simulated patterns. Dynamical patterns provide significantly more precision than kinematical patterns. Dynamical patterns also provide better estimates of tetragonality at low levels of distortion relative to the reference pattern; kinematical patterns can perform better at large values of relative tetragonality due to the ability to rapidly generate patterns relating to a distorted lattice. A library of dynamically generated patterns with different lattice parameters might be used to achieve a similar advantage. The convergence of the cross-correlation approach is also assessed for the different reference pattern types.

A Dictionary Approach to Electron Backscatter Diffraction Indexing.

We propose a framework for indexing of grain and subgrain structures in electron backscatter diffraction patterns of polycrystalline materials. We discretize the domain of a dynamical forward model onto a dense grid of orientations, producing a dictionary of patterns. For each measured pattern, we identify the most similar patterns in the dictionary, and identify boundaries, detect anomalies, and index crystal orientations. The statistical distribution of these closest matches is used in an unsupervised binary decision tree (DT) classifier to identify grain boundaries and anomalous regions. The DT classifies a pattern as an anomaly if it has an abnormally low similarity to any pattern in the dictionary. It classifies a pixel as being near a grain boundary if the highly ranked patterns in the dictionary differ significantly over the pixel's neighborhood. Indexing is accomplished by computing the mean orientation of the closest matches to each pattern. The mean orientation is estimated using a maximum likelihood approach that models the orientation distribution as a mixture of Von Mises-Fisher distributions over the quaternionic three sphere. The proposed dictionary matching approach permits segmentation, anomaly detection, and indexing to be performed in a unified manner with the additional benefit of uncertainty quantification.

Applications of the Clifford torus to material textures.

This paper introduces a new 2D representation of the orientation distribution function for an arbitrary material texture. The approach is based on the isometric square torus mapping of the Clifford torus, which allows for points on the unit quaternion hypersphere (each corresponding to a 3D orientation) to be represented in a periodic 2D square map. The combination of three such orthogonal mappings into a single RGB (red-green-blue) image provides a compact periodic representation of any set of orientations. Square torus representations of five different orientation sampling methods are compared and analyzed in terms of the Riesz s energies that quantify the uniformity of the samplings. The effect of crystallographic symmetry on the square torus map is analyzed in terms of the Rodrigues fundamental zones for the rotational symmetry groups. The paper concludes with example representations of important texture components in cubic and hexagonal materials. The new RGB representation provides a convenient and compact way of generating training data for the automated analysis of material textures by means of neural networks.

Dynamical electron backscatter diffraction patterns. Part I: pattern simulations.

A new approach for the simulation of dynamic electron backscatter diffraction (EBSD) patterns is introduced. The computational approach merges deterministic dynamic electron-scattering computations based on Bloch waves with a stochastic Monte Carlo (MC) simulation of the energy, depth, and directional distributions of the backscattered electrons (BSEs). An efficient numerical scheme is introduced, based on a modified Lambert projection, for the computation of the scintillator electron count as a function of the position and orientation of the EBSD detector; the approach allows for the rapid computation of an individual EBSD pattern by bi-linear interpolation of a master EBSD pattern. The master pattern stores the BSE yield as a function of the electron exit direction and exit energy and is used along with weight factors extracted from the MC simulation to obtain energy-weighted simulated EBSD patterns. Example simulations for nickel yield realistic patterns and energy-dependent trends in pattern blurring versus filter window energies are in agreement with experimental energy-filtered EBSD observations reported in the literature.

Orientation, pattern center refinement and deformation state extraction through global optimization algorithms.

Global optimization algorithms have been adopted to the simultaneously refinement of orientation and pattern center for electron backscatter diffraction patterns as well as deformation state extraction. The hyperparameter space and mutation schemes of differential evolution (DE) algorithm has been thoroughly investigated and showed to be a more efficient algorithm than the particle swarm optimization (PSO) algorithm. The optimal hyperparameters for DE generally depend on conditions such as the number of variables to be optimized and the size of bounded search space but reasonably close initial values for crossover probability is 0.9, mutation factor is 0.5, population size is ten times the number of variables, and number of iterations is at least 100. Validation on a set of simulated undeformed single crystal nickel patterns reveals a mean accuracy of ≈0.03° and ≈0.01% detector width across a large field of view. In addition, validation using noisy simulated deformed patterns with known deformation state and pattern center shows that the mean accuracy of shear strain and rotation components is ≈0.001 and for the normal strain ≈0.002.

Dynamic Symmetry Breaking in Chiral Magnetic Systems.

The Dzyaloshinskii-Moriya interaction (DMI) in magnetic systems stabilizes spin textures with preferred chirality, applicable to next-generation memory and computing architectures. In perpendicularly magnetized heavy-metal/ferromagnet films, the interfacial DMI originating from structural inversion asymmetry and strong spin-orbit coupling favors chiral Néel-type domain walls (DWs) whose energetics and mobility remain at issue. Here, a new effect is characterized in which domains expand unidirectionally in response to a combination of out-of-plane and in-plane magnetic fields, with the growth direction controlled by the in-plane field strength. These growth directionalities and symmetries with applied fields cannot be understood from static treatments alone. The authors theoretically demonstrate that perpendicular field torques stabilize steady-state magnetization profiles highly asymmetric in elastic energy, resulting in a dynamic symmetry breaking consistent with the experimental findings. This phenomenon sheds light on the mechanisms governing the dynamics of Néel-type DWs and expands the utility of field-driven DW motion to probe and control chiral DWs.

Electron backscattered diffraction using a new monolithic direct detector: High resolution and fast acquisition.

A monolithic active pixel sensor based direct detector that is optimized for the primary beam energies in scanning electron microscopes is implemented for electron back-scattered diffraction (EBSD) applications. The high detection efficiency of the detector and its large array of pixels allow sensitive and accurate detection of Kikuchi bands arising from primary electron beam excitation energies of 4 keV to 28 keV, with the optimal contrast occurring in the range of 8-16 keV. The diffraction pattern acquisition speed is substantially improved via a sparse sampling mode, resulting from the acquisition of a reduced number of pixels on the detector. Standard inpainting algorithms are implemented to effectively estimate the information in the skipped regions in the acquired diffraction pattern. For EBSD mapping, an acquisition speed as high as 5988 scan points per second is demonstrated, with a tolerable fraction of indexed points and accuracy. The collective capabilities spanning from high angular resolution EBSD patterns to high speed pattern acquisition are achieved on the same detector, facilitating simultaneous detection modalities that enable a multitude of advanced EBSD applications, including lattice strain mapping, structural refinement, low-dose characterization, 3D-EBSD and dynamic in situ EBSD.

Three-dimensional study of the vector potential of magnetic structures.

The vector potential is central to a number of areas of condensed matter physics, such as superconductivity and magnetism. We have used a combination of electron wave phase reconstruction and electron tomographic reconstruction to experimentally measure and visualize the three-dimensional vector potential in and around a magnetic Permalloy structure. The method can probe the vector potential of the patterned structures with a resolution of about 13 nm. A transmission electron microscope operated in the Lorentz mode is used to record four tomographic tilt series. Measurements for a square Permalloy structure with an internal closure domain configuration are presented.

EBSD pattern simulations for an interaction volume containing lattice defects.

A quantitative understanding of the effect of the spatial distribution and density of lattice defects on the electron backscatter diffraction patterns requires careful consideration of the electron-matter interaction volume and the traction free boundary condition on the deformation field for near-surface defects. In this work, we couple a depth-specific dynamical electron scattering simulation with an approximate crystal deformation model to generate a single diffraction pattern from an interaction volume containing lattice defects. Two case studies are considered, namely a single edge dislocation and a low angle grain boundary. Their displacement fields, derived from the three-dimensional Yoffe-Shaibani-Hazzeldine's dislocation field model, are fed into the simulation and the resulting diffraction patterns are cross-validated using the HR-EBSD technique. In addition, diffraction contrast associated with defect deformation field is investigated with the virtual beam technique. Pattern diffuseness is quantitatively analyzed in the frequency domain as a function of dislocation density.

High resolution low kV EBSD of heavily deformed and nanocrystalline Aluminium by dictionary-based indexing.

We demonstrate the capability of a novel Electron Backscatter Diffraction (EBSD) dictionary indexing (DI) approach by means of orientation mapping of a highly deformed graded microstructure in a shot peened Aluminium 7075-T651 alloy. A low microscope accelerating voltage was used to extract, for the first time from a bulk sample, statistically significant orientation information from a region close to a shot crater, showing both recrystallized nano-grains and heavily deformed grains. We show that the robust nature of the DI method allows for faster acquisition of lower quality patterns, limited only by the camera hardware, compared to the acquisition speed and pattern quality required for the conventional Hough indexing (HI) approach. The proposed method paves the way for the quantitative and accurate EBSD characterization of heavily deformed microstructures at a sub-micrometer length scale in cases where the current indexing techniques largely fail.

Transmission scanning electron microscopy: Defect observations and image simulations.

The new capabilities of a FEG scanning electron microscope (SEM) equipped with a scanning transmission electron microscopy (STEM) detector for defect characterization have been studied in parallel with transmission electron microscopy (TEM) imaging. Stacking faults and dislocations have been characterized in strontium titanate, a polycrystalline nickel-base superalloy and a single crystal cobalt-base material. Imaging modes that are similar to conventional TEM (CTEM) bright field (BF) and dark field (DF) and STEM are explored, and some of the differences due to the different accelerating voltages highlighted. Defect images have been simulated for the transmission scanning electron microscopy (TSEM) configuration using a scattering matrix formulation, and diffraction contrast in the SEM is discussed in comparison to TEM. Interference effects associated with conventional TEM, such as thickness fringes and bending contours are significantly reduced in TSEM by using a convergent probe, similar to a STEM imaging modality, enabling individual defects to be imaged clearly even in high dislocation density regions. Beyond this, TSEM provides significant advantages for high throughput and dynamic in-situ characterization.

Magnetic phase shift reconstruction for uniformly magnetized nanowires.

A new analytical model is developed for the magnetic phase shift of uniformly magnetized nanowires with ideal cylindrical geometry. The model is applied to experimental data from off-axis electron holography measurements of the phase shift of CoFeB nanowires, and the saturation induction of a selected wire, as well as its radius, aspect ratio, position and orientation, is determined by fitting the model parameters. The saturation induction value of 1.7T of the CoFeB nanowire is found to be similar, to be within the measurement error, to values reported in the literature.