Resolution of Virtual Depth Sectioning from Four-Dimensional Scanning Transmission Electron Microscopy.

One approach to three-dimensional structure determination using the wealth of scattering data in four-dimensional (4D) scanning transmission electron microscopy (STEM) is the parallax method proposed by Ophus et al. (2019. Advanced phase reconstruction methods enabled by 4D scanning transmission electron microscopy, Microsc Microanal25, 10-11), which determines the scattering matrix and uses it to synthesize a virtual depth-sectioning reconstruction of the sample structure. Drawing on an equivalence with a hypothetical confocal imaging mode, we derive contrast transfer and point spread functions for this parallax method applied to weakly scattering objects, showing them identical to earlier depth-sectioning STEM modes when only bright field signal is used, but that improved depth resolution is possible if dark field signal can be used. Through a simulation-based study of doped Si, we show that this depth resolution is preserved for thicker samples, explore the impact of shot noise on the parallax reconstructions, discuss challenges to making use of dark field signal, and identify cases where the interpretation of the parallax reconstruction breaks down.

Braiding and Fusion of Non-Abelian Vortex Anyons.

We have studied topology and dynamics of quantum vortices in spin-2 Bose-Einstein condensates. By computationally modeling controllable braiding and fusion of these vortices, we have demonstrated that certain vortices in such spinor condensates behave as non-Abelian anyons. We identify these anyons as fluxon, chargeon, and dyon quasiparticles. The pertinent anyon models are defined by the quantum double of the underlying discrete non-Abelian symmetry group of the condensate ground state order parameter.

Simple Wave-Optical Superpositions as Prime Number Sieves.

We encode the sequence of prime numbers into simple superpositions of identical waves, mimicking the archetypal prime number sieve of Eratosthenes. The primes are identified as zeros accompanied by phase singularities in a physically generated wave field for integer valued momenta. Similarly, primes are encoded in the diffraction pattern from a simple single aperture and in the harmonics of a single vibrating resonator. Further, diffraction physics connections to number theory reveal how to encode all Gaussian primes, twin primes, and how to construct wave fields with amplitudes equal to the divisor function at integer spatial frequencies. Remarkably, all of these basic diffraction phenomena reveal that the naturally irregular sequence of primes can arise from trivially ordered wave superpositions.

Vacancy induced formation of nanoporous silicon, carbon and silicon carbide.

Nanoporous semiconductors are used in a range of applications from sensing and gas separation, to photovoltaics, rechargeable batteries, energetic materials and micro electro mechanical systems. In most cases porosity occurs in conjunction with the competing process of amorphisation, creating a complicated material that responds differently to strain and density changes, depending on the composition. In this paper we use simple computational workflow involving Monte Carlo simulation, numerical characterisation and statistical analysis to explore the development of amorphous and nanoporous carbon, silicon and silicon carbide. We show that amorphous regions in Si and SiC form in advance of nanopores, and are essential in stabilising the nanopores once developed. Carbon prefers a porous structure at lower strains than amorphisation and exhibits a bimodal change in the structure which correlates with the change in C-C bond angles from tetrahedral sp3-like bonds to hexagonal sp2-like bonds as the strain increases. These results highlight how both of these processes can be analysed simultaneously using reliable interatomic forcefields or density functionals, provided sufficient samples are included to support the statistics.

Effect of specimen orientation on the accuracy of vector field electron tomography.

Vector field electron tomography (VFET) reconstructs vector fields based on phase maps recorded from two or more orthogonal tilt series. The tomographic reconstruction of vector fields involves considerations beyond those involved in the reconstruction of scalar fields. Here we examine the effect of initial magnetization orientation on reconstruction errors. The orientation of a magnetic particle affects the contrast in the phase maps. This, in turn, affects the accuracy of the reconstructed vector fields. We derive expressions that model the dependence of reconstruction errors on initial specimen orientation when using a filtered backprojection algorithm to reconstruct a vector potential from two tilt series. We compare these analytical results with those from numerical simulations. Our results can inform experimental procedures, such as sample preparation techniques and the choice of tilt series orientations. Specimen orientation can be a significant source of error in VFET, and our results can provide the means to minimize these errors.

Publisher's Note: Calculation of Projected Bond-Orientational Order Parameters to Quantify Local Symmetries from Transmission Diffraction Data [Phys. Rev. Lett. 116, 205501 (2016)].

This corrects the article DOI: 10.1103/PhysRevLett.116.205501.

Calculation of Projected Bond-Orientational Order Parameters to Quantify Local Symmetries from Transmission Diffraction Data.

The bond-orientational order parameters introduced by Steinhardt et al. [Phys. Rev. B 28, 784 (1983)] have been an invaluable measurement tool for assessing short-range order in disordered, close-packed assemblies of particles in which the particle positions are known. In many glassy systems the measurement of particle position is not possible or limited (field of view, thickness, resolution) and the bond-orientational order parameters cannot be measured, or adequately sampled. Here we calculate a set of rotationally averaged, projected bond-orientational order parameters that reflect the symmetries of close-packed particle clusters when projected onto a plane. We show by simulation that these parameters are unique fingerprints that can be directly compared to angular correlations in limited-volume, transmission geometry, diffraction patterns from close-packed glassy assemblies.

Electron vortex production and control using aberration induced diffraction catastrophes.

An aberration corrected electron microscope is used to create electron diffraction catastrophes, containing arrays of intensity zeros threading vortex cores. Vortices are ascribed to these arrays using catastrophe theory, scalar diffraction integrals, and experimentally retrieved phase maps. From measured wave function phases, obtained using focal-series phase retrieval, the orbital angular momentum density is mapped for highly astigmatic electron probes. We observe vortex rings and topological reconnections of nodal lines by tracking the vortex cores using the retrieved phases.

Volume imaging by tracking sparse topological features in electron micrograph tilt series.

The sensitive coherent interference of electron waves arising from a specimen is useful for revealing subtle structural information in electron micrographs, which can be important for minimising dose and for rapid imaging. In general, dynamical diffraction is expected due to the useful strong interactions of electrons with matter, which can create phase contrast that violates the requisite Radon projection assumption for tomography. It is for these reasons that incoherent imaging modalities such as high angle annular dark field have been favoured to date in electron tomography of crystalline specimens, to access a monotonic relationship between specimen thickness and micrograph intensity. Here we use a geometric approach to track topological features that are robust to perturbation of the imaging conditions, to enable 3D reconstructions from electron microscope tilt series under imaging conditions that violate the Radon projection assumption, with an emphasis on phase contrast. Invoking a sparsity assumption, we demonstrate that topological features can be reliably tracked in 3D using a differential geometric form of stereoscopy, to circumvent departures from the projection approximation and reduce noise by effecting segmentation of interest points from the outset. We demonstrate this approach on a variety of different specimen and data types, from polyhedral nanoparticles, to steel dislocation networks, cryo-EM cellular structures and 3D diffuse diffraction of a relaxor ferroelectric.

Factors limiting quantitative phase retrieval in atomic-resolution differential phase contrast scanning transmission electron microscopy using a segmented detector.

Quantitative differential phase contrast imaging of materials in atomic-resolution scanning transmission electron microscopy using segmented detectors is limited by various factors, including coherent and incoherent aberrations, detector positioning and uniformity, and scan-distortion. By comparing experimental case studies of monolayer and few-layer graphene with image simulations, we explore which parameters require the most precise characterisation for reliable and quantitative interpretation of the reconstructed phases. Coherent and incoherent lens aberrations are found to have the most significant impact. For images over a large field of view, the impact of noise and non-periodic boundary conditions are appreciable, but in this case study have less of an impact than artefacts introduced by beam deflections coupling to beam scanning (imperfect tilt-shift purity).

Suppressing dynamical diffraction artefacts in differential phase contrast scanning transmission electron microscopy of long-range electromagnetic fields via precession.

It is well known that dynamical diffraction varies with changes in sample thickness and local crystal orientation (due to sample bending). In differential phase contrast scanning transmission electron microscopy (DPC-STEM), this can produce contrast comparable to that arising from the long-range electromagnetic fields probed by this technique. Through simulation we explore the scale of these dynamical diffraction artefacts and introduce a metric for the magnitude of their contribution to the contrast. We show that precession over an angular range of a few milliradian can suppress this contribution to the contrast by one-to-two orders of magnitude. Our exploration centres around a case study of GaAs near the [011] zone-axis orientation using a probe-forming aperture semiangle on the order of 0.1 mrad at 300 keV, but the trends found and methodology used are expected to apply more generally.

High contrast at low dose using a single, defocussed transmission electron micrograph.

For many soft-matter specimens, transmission electron microscopists face the double-bind of low contrast images, due to weakly-scattering specimens, alongside severe limits on the electron dose that can be used before the specimen is damaged by the electron beam. The combination of these effects causes the resultant micrographs to have very low signal-to-noise. It is well known that varying the defocus aberration can enhance image contrast in electron microscopy. For single-material objects where the variation of absorption and phase contrast are functions of one another, since both are governed by the variation in thickness profile, we show that the thickness profile can be reconstructed at very low dose. The algorithm, first established in X-ray imaging, requires some a priori information but only a single defocussed image of the region of interest, making it more dose efficient than either a conventional transport-of-intensity phase reconstruction (which would require two images and tends to amplify noise), or an absorption-contrast analysis of a single in-focus image recorded at the same electron dose (which does not benefit from the significant signal-to-noise enhancement of the present algorithm). These findings are presented through both simulations and experimental data.

Measuring nanometre-scale electric fields in scanning transmission electron microscopy using segmented detectors.

Electric field mapping using segmented detectors in the scanning transmission electron microscope has recently been achieved at the nanometre scale. However, converting these results to quantitative field measurements involves assumptions whose validity is unclear for thick specimens. We consider three approaches to quantitative reconstruction of the projected electric potential using segmented detectors: a segmented detector approximation to differential phase contrast and two variants on ptychographical reconstruction. Limitations to these approaches are also studied, particularly errors arising from detector segment size, inelastic scattering, and non-periodic boundary conditions. A simple calibration experiment is described which corrects the differential phase contrast reconstruction to give reliable quantitative results despite the finite detector segment size and the effects of plasmon scattering in thick specimens. A plasmon scattering correction to the segmented detector ptychography approaches is also given. Avoiding the imposition of periodic boundary conditions on the reconstructed projected electric potential leads to more realistic reconstructions.

Refinements in the collection of energy filtered diffraction patterns from disordered materials.

In this paper a method for collecting electron diffraction patterns using a Gatan imaging filter is presented. The method enables high-quality diffraction data to be measured at scattering angles comparable to those that can be obtained using X-ray and neutron diffraction. In addition, the method offers the capability for examining small regions of sample in, for example, thin films and nano-structures. Using X-ray, neutron and electron diffraction data collected from the same sample, we demonstrate quantitative agreement between all three. We also present a novel method for obtaining the single scattering contribution to the total diffracted intensity by collecting data at various electron wavelengths. This approach allows pair distribution functions to be determined from electron diffraction in cases where there exists significant multiple scattering.

RDFTools: a software tool for quantifying short-range ordering in amorphous materials.

A software package for computing radial distribution functions and other pair correlation functions from electron diffraction patterns of disordered solids is presented. The package, called RDFTools, is freely available via the internet and allows rapid in situ measurements of such quantities as interatomic nearest neighbor distances, average bond angles and coordination numbers. The software runs under DigitalMicrograph™ (Pleasanton, California, Gatan), a very widely used program in transmission electron microscopy. All implemented algorithms have been designed to compute diffraction integrals and data-processing averages in a fast and efficient manner to enable quick processing of publication ready, quantitative pair distribution function information. In the development of RDFTools, significant attention was paid to provide a robust and intuitive user-interface for deriving reliable semiquantitative information. For example, RDFTools enables accurate pair separation distances to be revealed upon immediate interrogation at the microscope; even for potentially thick specimens and/or regions of unknown elemental composition.

Macroscopic electrical field distribution and field-induced surface stresses of needle-shaped field emitters.

One major concern since the development of the field ion microscope is the mechanical strength of the specimens. The macroscopic shape of the imaging tip greatly influences field-induced stresses and there is merit in further study of this phenomenon from a classical perspective. Understanding the geometrical, as opposed to localized electronic, factors that affect the stress might improve the quality and success rate of atom probe experiments. This study uses macroscopic electrostatic principles and finite element modelling to investigate field-induced stresses in relation to the shape of the tip. Three two-dimensional idealized models are considered, namely hyperbolic, parabolic and sphere-on-orthogonal-cone; the shapes of which are compared to experimental tips prepared by electro-polishing. Three dimensional morphologies of both a nano-porous and single-crystal aluminium tip are measured using electron tomography to quantitatively test the assumption of cylindrical symmetry for electro-polished tips. The porous tip was prepared and studied to demonstrate a fragile specimen for which such finite element studies could determine potential mechanical failure, prior to any exhaustive atom probe investigation.

Modeling of structure and porosity in amorphous silicon systems using Monte Carlo methods.

Porous solids are very important from a scientific point of view as they provide a medium in which to study the behavior of confined fluids. Although some porous solids have a well defined pore geometry such as zeolites, many porous solids lack crystalline order and are usually described as amorphous. The description of the pore geometry in such structures is very difficult. The authors develop a modeling approach using a Monte Carlo algorithm to simulate porosity within amorphous systems based on constraints for the internal volume and surface area. To illustrate this approach, a model of microporous amorphous silicon is presented. Structural aspects of the porous model are then compared against hybrid reverse Monte Carlo simulations of nonporous amorphous silicon and published results from the literature. It is found that coordination defects are predominately located at the pore surface walls.