Direct imaging of electron density with a scanning transmission electron microscope

Recent studies of secondary electron (SE) emission in scanning transmission electron microscopes suggest that material's properties such as electrical conductivity, connectivity, and work function can be probed with atomic scale resolution using a technique known as secondary electron e-beam-induced current (SEEBIC). Here, we apply the SEEBIC imaging technique to a stacked 2D heterostructure device to reveal the spatially resolved electron density of an encapsulated WSe2 layer. We find that the double Se lattice site shows higher emission than the W site, which is at odds with first-principles modelling of valence ionization of an isolated WSe2 cluster. These results illustrate that atomic level SEEBIC contrast within a single material is possible and that an enhanced understanding of atomic scale SE emission is required to account for the observed contrast. In turn, this suggests that, in the future, subtle information about interlayer bonding and the effect on electron orbitals could be directly revealed with this technique.

Electron-Beam Induced Emergence of Mesoscopic Ordering in Layered MnPS3

Ordered mesoscale structures in 2D materials induced by small misorientations have allowed for a wide variety of electronic, ferroelectric, and quantum phenomena to be explored. Until now, the only mechanism to induce this periodic ordering was via mechanical rotations between the layers, with the periodicity of the resulting moiré pattern being directly related to twist angle. Here we report a fundamentally distinct mechanism for emergence of mesoscopic periodic patterns in multilayer sulfur-containing metal phosphorus trichalcogenide, MnPS₃, induced by the electron beam. The formation under the beam of periodic hexagonal patterns with several characteristic length scales, nucleation and transitions between the phases, and local dynamics are demonstrated. The associated mechanisms are attributed to the relative contraction of the layers caused by beam-induced sulfur vacancy formation with subsequent ordering and lattice parameter change. As a result, the plasmonic response of the system is locally altered, suggesting an element of control over plasmon resonances by electron beam patterning. We pose that harnessing this phenomenon provides both insight into fundamental physics of quantum materials and enables device applications by enabling controlled periodic potentials on the atomic scale.

Automated Experiment in 4D-STEM: Exploring Emergent Physics and Structural Behaviors

Automated experiments in 4D scanning transmission electron microscopy (STEM) are implemented for rapid discovery of local structures, symmetry-breaking distortions, and internal electric and magnetic fields in complex materials. Deep kernel learning enables active learning of the relationship between local structure and 4D-STEM-based descriptors. With this, efficient and "intelligent" probing of dissimilar structural elements to discover desired physical functionality is made possible. This approach allows effective navigation of the sample in an automated fashion guided by either a predetermined physical phenomenon, such as strongest electric field magnitude, or in an exploratory fashion. We verify the approach first on preacquired 4D-STEM data and further implement it experimentally on an operational STEM. The experimental discovery workflow is demonstrated using graphene and subsequently extended toward a lesser-known layered 2D van der Waals material, MnPS₃. This approach establishes a pathway for physics-driven automated 4D-STEM experiments that enable probing the physics of strongly correlated systems and quantum materials and devices, as well as exploration of beam-sensitive materials.

The importance of temporal and spatial incoherence in quantitative interpretation of 4D-STEM

Recent developments in pixelated detectors, when combined with aberration correction of probe forming optics have greatly enhanced the field of scanning electron diffraction. Differential phase contrast is now routine and deep learning has been proposed as a method to extract maximum information from diffraction patterns. This work examines the effects of temporal and spatial incoherence on convergent beam electron diffraction patterns and demonstrates that simple center of mass measurements cannot be naively interpreted. The inclusion of incoherence in deep learning data sets is also discussed.

Structure retrieval from four-dimensional scanning transmission electron microscopy: Statistical analysis of potential pitfalls in high-dimensional data

Four-dimensional (4D) scanning transmission electron microscopy is one of the most rapidly growing modes of electron microscopy imaging. The advent of fast pixelated cameras and the associated data infrastructure have greatly accelerated this process. Yet conversion of the 4D datasets into physically meaningful structure images in real space remains an open issue. In this work, we demonstrate that it is possible to systematically create filters that will affect the apparent resolution or even qualitative features of the real-space structure image, reconstructing artificially generated patterns. As initial efforts, we explore statistical model selection algorithms, aiming for robustness and reliability of estimated filters. This statistical model selection analysis demonstrates the need for regularization and cross validation of inversion methods to robustly recover structure from high-dimensional diffraction datasets.

Direct Cation Exchange in Monolayer MoS_{2} via Recombination-Enhanced Migration

In addition to their unique optical and electronic properties, two-dimensional materials provide opportunities to directly observe atomic-scale defect dynamics. Here we use scanning transmission electron microscopy to observe substitutional Re impurities in monolayer MoS_{2} undergo direct exchanges with neighboring Mo atoms in the lattice. Density-functional-theory calculations find that the energy barrier for direct exchange, a process that has only been studied as a diffusion mechanism in bulk materials, is too large for either thermal activation or energy directly transferred from the electron beam. The presence of multiple sulfur vacancies next to the exchanged Re-Mo pair, as observed by electron microscopy, does not lower the energy barrier sufficiently to account for the observed atomic exchange. Instead, the calculations find that a Re dopant and surrounding sulfur vacancies introduce an ever-changing set of deep levels in the energy gap. We propose that these levels mediate an "explosive" recombination-enhanced migration via multiple electron-hole recombination events. As a proof of concept, we also show that Re-Mo direct exchange can be triggered via controlled creation of sulfur vacancies. The present experimental and theoretical findings lay a fundamental framework towards manipulating single substitutional dopants in two-dimensional materials.

Ultra-high resolution electron microscopy

The last two decades have seen dramatic advances in the resolution of the electron microscope brought about by the successful correction of lens aberrations that previously limited resolution for most of its history. We briefly review these advances, the achievement of sub-Ångstrom resolution and the ability to identify individual atoms, their bonding configurations and even their dynamics and diffusion pathways. We then present a review of the basic physics of electron scattering, lens aberrations and their correction, and an approximate imaging theory for thin crystals which provides physical insight into the various different imaging modes. Then we proceed to describe a more exact imaging theory starting from Yoshioka's formulation and covering full image simulation methods using Bloch waves, the multislice formulation and the frozen phonon/quantum excitation of phonons models. Delocalization of inelastic scattering has become an important limiting factor at atomic resolution. We therefore discuss this issue extensively, showing how the full-width-half-maximum is the appropriate measure for predicting image contrast, but the diameter containing 50% of the excitation is an important measure of the range of the interaction. These two measures can differ by a factor of 5, are not a simple function of binding energy, and full image simulations are required to match to experiment. The Z-dependence of annular dark field images is also discussed extensively, both for single atoms and for crystals, and we show that temporal incoherence must be included accurately if atomic species are to be identified through matching experimental intensities to simulations. Finally we mention a few promising directions for future investigation.

Atomic-Level Sculpting of Crystalline Oxides: Toward Bulk Nanofabrication with Single Atomic Plane Precision

The atomic-level sculpting of 3D crystalline oxide nanostructures from metastable amorphous films in a scanning transmission electron microscope (STEM) is demonstrated. Strontium titanate nanostructures grow epitaxially from the crystalline substrate following the beam path. This method can be used for fabricating crystalline structures as small as 1-2 nm and the process can be observed in situ with atomic resolution. The fabrication of arbitrary shape structures via control of the position and scan speed of the electron beam is further demonstrated. Combined with broad availability of the atomic resolved electron microscopy platforms, these observations suggest the feasibility of large scale implementation of bulk atomic-level fabrication as a new enabling tool of nanoscience and technology, providing a bottom-up, atomic-level complement to 3D printing.

Local observation of the site occupancy of Mn in a MnFePSi compound

MnFePSi compounds are promising materials for magnetic refrigeration as they exhibit a giant magnetocaloric effect. From first principles calculations and experiments on bulk materials, it has been proposed that this is due to the Mn and Fe atoms preferentially occupying two different sites within the atomic lattice. A recently developed technique was used to deconvolve the obscuring effects of both multiple elastic scattering and thermal diffuse scattering of the probe in an atomic resolution electron energy-loss spectroscopy investigation of a MnFePSi compound. This reveals, unambiguously, that the Mn atoms preferentially occupy the 3g site in a hexagonal crystal structure, confirming the theoretical predictions. After deconvolution, the data exhibit a difference in the Fe L_{2,3} ratio between the 3f and 3g sites consistent with differences in magnetic moments calculated from first principles, which are also not observed in the raw data.

Atomic scale studies of La/Sr ordering in colossal magnetoresistant La(2-2x)Sr(1+2x)Mn2O7 single crystals

To date, it is unclear whether chemical order (or disorder) is in any way connected to double exchange, electronic phase separation, or charge ordering (CO) in manganites. In this work, we carry out an atomic resolution study of the colossal magnetoresistant manganite La(2-2x)Sr(1+2x)Mn2O7 (LSMO). We combine aberration-corrected electron microscopy and spectroscopy with spectroscopic image simulations, to analyze cation ordering at the atomic scale in real space in a number of LSMO single crystals. We compare three different compositions within the phase diagram: a ferromagnetic metallic material (x=0.36), an insulating, antiferromagnetic charge ordered (AF-CO) compound (x=0.5), which also exhibits orbital ordering, and an additional AF sample (x=0.56). Detailed image simulations are essential to accurately quantify the degree of chemical ordering of these samples. We find a significant degree of long-range chemical ordering in all cases, which increases in the AF-CO range. However, the degree of ordering is never complete nor can it explain the strongly correlated underlying ordering phenomena. Our results show that chemical ordering over distinct crystallographic sites is not needed for electronic ordering phenomena to appear in manganites, and cannot by itself explain the complex electronic behavior of LSMO.

Direct observation of ferroelectric field effect and vacancy-controlled screening at the BiFeO3/LaxSr1-xMnO3 interface

The development of interface-based magnetoelectric devices necessitates an understanding of polarization-mediated electronic phenomena and atomistic polarization screening mechanisms. In this work, the LSMO/BFO interface is studied on a single unit-cell level through a combination of direct order parameter mapping by scanning transmission electron microscopy and electron energy-loss spectroscopy. We demonstrate an unexpected ~5% lattice expansion for regions with negative polarization charge, with a concurrent anomalous decrease of the Mn valence and change in oxygen K-edge intensity. We interpret this behaviour as direct evidence for screening by oxygen vacancies. The vacancies are predominantly accumulated at the second atomic layer of BFO, reflecting the difference of ionic conductivity between the components. This vacancy exclusion from the interface leads to the formation of a tail-to-tail domain wall. At the same time, purely electronic screening is realized for positive polarization charge, with insignificant changes in lattice and electronic properties. These results underline the non-trivial role of electrochemical phenomena in determining the functional properties of oxide interfaces. Furthermore, these behaviours suggest that vacancy dynamics and exclusion play major roles in determining interface functionality in oxide multilayers, providing clear implications for novel functionalities in potential electronic devices.

Simulation of probe position-dependent electron energy-loss fine structure

We present a theoretical framework for calculating probe-position-dependent electron energy-loss near-edge structure for the scanning transmission electron microscope by combining density functional theory with dynamical scattering theory. We show how simpler approaches to calculating near-edge structure fail to include the fundamental physics needed to understand the evolution of near-edge structure as a function of probe position and investigate the dependence of near-edge structure on probe size. It is within this framework that density functional theory should be presented, in order to ensure that variations of near-edge structure are truly due to local electronic structure and how much from the diffraction and focusing of the electron beam.

Grain-boundary-enhanced carrier collection in CdTe solar cells

When CdTe solar cells are doped with Cl, the grain boundaries no longer act as recombination centers but actively contribute to carrier collection efficiency. The physical origin of this remarkable effect has been determined through a combination of aberration-corrected scanning transmission electron microscopy, electron energy loss spectroscopy, and first-principles theory. Cl substitutes for a large proportion of the Te atoms within a few unit cells of the grain boundaries. Density functional calculations reveal the mechanism, and further indicate the grain boundaries are inverted to n type, establishing local p-n junctions which assist electron-hole pair separation. The mechanism is electrostatic, and hence independent of the geometry of the boundary, thereby explaining the universally high collection efficiency of Cl-doped CdTe solar cells.

Simulation of spatially resolved electron energy loss near-edge structure for scanning transmission electron microscopy

Aberration-corrected scanning transmission electron microscopy yields probe-position-dependent energy-loss near-edge structure (ELNES) measurements, potentially providing spatial mapping of the underlying electronic states. ELNES calculations, however, typically describe excitations by a plane wave traveling in vacuum, neglecting the interaction of the electron probe with the local electronic environment as it propagates through the specimen. Here, we report a methodology that combines a full electronic-structure calculation with propagation of a focused beam in a thin film. The results demonstrate that only a detailed calculation using this approach can provide quantitative agreement with observed variations in probe-position-dependent ELNES.

Single atom microscopy

We show that aberration-corrected scanning transmission electron microscopy operating at low accelerating voltages is able to analyze, simultaneously and with single atom resolution and sensitivity, the local atomic configuration, chemical identities, and optical response at point defect sites in monolayer graphene. Sequential fast-scan annular dark-field (ADF) imaging provides direct visualization of point defect diffusion within the graphene lattice, with all atoms clearly resolved and identified via quantitative image analysis. Summing multiple ADF frames of stationary defects produce images with minimized statistical noise and reduced distortions of atomic positions. Electron energy-loss spectrum imaging of single atoms allows the delocalization of inelastic scattering to be quantified, and full quantum mechanical calculations are able to describe the delocalization effect with good accuracy. These capabilities open new opportunities to probe the defect structure, defect dynamics, and local optical properties in 2D materials with single atom sensitivity.

Exploring mesoscopic physics of vacancy-ordered systems through atomic scale observations of topological defects

Vacancy-ordered transition metal oxides have multiple similarities to classical ferroic systems including ferroelectrics and ferroelastics. The expansion coefficients for corresponding Ginzburg-Landau-type free energies are readily accessible from bulk phase diagrams. Here, we demonstrate that the gradient and interfacial terms can quantitatively be determined from the atomically resolved scanning transmission electron microscopy data of the topological defects and interfaces in model lanthanum-strontium cobaltite. With this knowledge, the interplay between ordering, chemical composition, and mechanical effects at domain walls, interfaces and structural defects can be analyzed.

Electron transfer and ionic displacements at the origin of the 2D electron gas at the LAO/STO interface: direct measurements with atomic-column spatial resolution

Using state-of-the-art, aberration-corrected scanning transmission electron microscopy and electron energy loss spectroscopy with atomic-scale spatial resolution, experimental evidence for an intrinsic electronic reconstruction at the LAO/STO interface is shown. Simultaneous measurements of interfacial electron density and system polarization are crucial for establishing the highly debated origin of the 2D electron gas.

Atomic-resolution imaging of spin-state superlattices in nanopockets within cobaltite thin films

Certain cobalt oxides are known to exhibit ordered Co spin states, as determined from macroscopic techniques. Here we report real-space atomic-resolution imaging of Co spin-state ordering in nanopockets of La(0.5)Sr(0.5)CoO(3-δ) thin films. Unlike the bulk material, where no Co spin-state ordering is found, thin films present a strain-induced domain structure due to oxygen vacancy ordering, inside of which some nanometer sized domains show high-spin Co ions in the planes containing O vacancies and low-spin Co ions in the stoichiometric planes. First-principles calculations provide support for this interpretation.

Atomic-scale compensation phenomena at polar interfaces

The interfacial screening charge that arises to compensate electric fields of dielectric or ferroelectric thin films is now recognized as the most important factor in determining the capacitance or polarization of ultrathin ferroelectrics. Here we investigate using aberration-corrected electron microscopy and density-functional theory to show how interfaces cope with the need to terminate ferroelectric polarization. In one case, we show evidence for ionic screening, which has been predicted by theory but never observed. For a ferroelectric film on an insulating substrate, we found that compensation can be mediated by an interfacial charge generated, for example, by oxygen vacancies.

Mapping octahedral tilts and polarization across a domain wall in BiFeO3 from Z-contrast scanning transmission electron microscopy image atomic column shape analysis

Oxygen octahedral tilts underpin the functionality of a large number of perovskite-based materials and heterostructures with competing order parameters. We show how a precise analysis of atomic column shapes in Z-contrast scanning transmission electron microscopy images can reveal polarization and octahedral tilt behavior across uncharged and charged domain walls in BiFeO(3). This method is capable of visualizing octahedral tilts to much higher thicknesses than phase contrast imaging. We find that the octahedral tilt transition across a charged domain wall is atomically abrupt, while the associated polarization profile is diffuse (1.5-2 nm). Ginzburg-Landau theory then allows the relative contributions of polarization and the structural order parameters to the wall energy to be determined.

Suppression of octahedral tilts and associated changes in electronic properties at epitaxial oxide heterostructure interfaces

Epitaxial oxide interfaces with broken translational symmetry have emerged as a central paradigm behind the novel behaviors of oxide superlattices. Here, we use scanning transmission electron microscopy to demonstrate a direct, quantitative unit-cell-by-unit-cell mapping of lattice parameters and oxygen octahedral rotations across the BiFeO3-La0.7 Sr0.3 MnO3 interface to elucidate how the change of crystal symmetry is accommodated. Combined with low-loss electron energy loss spectroscopy imaging, we demonstrate a mesoscopic antiferrodistortive phase transition near the interface in BiFeO3 and elucidate associated changes in electronic properties in a thin layer directly adjacent to the interface.

Aberration-corrected scanning transmission electron microscopy: from atomic imaging and analysis to solving energy problems

The new possibilities of aberration-corrected scanning transmission electron microscopy (STEM) extend far beyond the factor of 2 or more in lateral resolution that was the original motivation. The smaller probe also gives enhanced single atom sensitivity, both for imaging and for spectroscopy, enabling light elements to be detected in a Z-contrast image and giving much improved phase contrast imaging using the bright field detector with pixel-by-pixel correlation with the Z-contrast image. Furthermore, the increased probe-forming aperture brings significant depth sensitivity and the possibility of optical sectioning to extract information in three dimensions. This paper reviews these recent advances with reference to several applications of relevance to energy, the origin of the low-temperature catalytic activity of nanophase Au, the nucleation and growth of semiconducting nanowires, and the origin of the eight orders of magnitude increased ionic conductivity in oxide superlattices. Possible future directions of aberration-corrected STEM for solving energy problems are outlined.

Atomic-resolution spectroscopic imaging: past, present and future

This review examines the development of atomically resolved electron energy loss spectroscopy from the first demonstration of plane-by-plane compositional profiling, through column-by-column spectroscopy to full two-dimensional and potentially three-dimensional spectroscopic imaging. Examples will be presented to highlight the increasing analytical sensitivity and image contrast obtained through each generation of aberration correction, moving towards the ultimate goal of mapping electronic structure inside materials with atomic resolution.

Spatial resolution and information transfer in scanning transmission electron microscopy

The relation between image resolution and information transfer is explored. It is shown that the existence of higher frequency transfer in the image is just a necessary but not sufficient condition for the achievement of higher resolution. Adopting a two-point resolution criterion, we suggest that a 10% contrast level between two features in an image should be used as a practical definition of resolution. In the context of scanning transmission electron microscopy, it is shown that the channeling effect does not have a direct connection with image resolution because sharp channeling peaks do not move with the scanning probe. Through a quantitative comparison between experimental image and simulation, a Fourier-space approach is proposed to estimate defocus and sample thickness. The effective atom size in Z-contrast imaging depends on the annular detector's inner angle. Therefore, an optimum angle exists for the highest resolution as a trade-off between reduced atom size and reduced signal with limited information transfer due to noise.

Modeling atomic-resolution scanning transmission electron microscopy images

A real-space description of inelastic scattering in scanning transmission electron microscopy is derived with particular attention given to the implementation of the projected potential approximation. A hierarchy of approximations to expressions for inelastic images is presented. Emphasis is placed on the conditions that must hold in each case. The expressions that justify the most direct, visual interpretation of experimental data are also the most approximate. Therefore, caution must be exercised in selecting experimental parameters that validate the approximations needed for the analysis technique used. To make the most direct, visual interpretation of electron-energy-loss spectroscopic images from core-shell excitations requires detector improvements commensurate with those that aberration correction provides for the probe-forming lens. Such conditions can be relaxed when detailed simulations are performed as part of the analysis of experimental data.

Depth sectioning in scanning transmission electron microscopy based on core-loss spectroscopy

Recent and ongoing improvements in aberration correction have opened up the possibility of depth sectioning samples using the scanning transmission electron microscope in a fashion similar to the confocal scanning optical microscope. We explore questions of principle relating to image interpretability in the depth sectioning of samples using electron energy loss spectroscopy. We show that provided electron microscope probes are sufficiently fine and detector collection semi-angles are sufficiently large we can expect to locate dopant atoms inside a crystal. Furthermore, unlike high angle annular dark field imaging, electron energy loss spectroscopy can resolve dopants of smaller atomic mass than the supporting crystalline matrix.

Modelling high-resolution electron microscopy based on core-loss spectroscopy

There are a number of factors affecting the formation of images based on core-loss spectroscopy in high-resolution electron microscopy. We demonstrate unambiguously the need to use a full nonlocal description of the effective core-loss interaction for experimental results obtained from high angular resolution electron channelling electron spectroscopy. The implications of this model are investigated for atomic resolution scanning transmission electron microscopy. Simulations are used to demonstrate that core-loss spectroscopy images formed using fine probes proposed for future microscopes can result in images that do not correspond visually with the structure that has led to their formation. In this context, we also examine the effect of varying detector geometries. The importance of the contribution to core-loss spectroscopy images by dechannelled or diffusely scattered electrons is reiterated here.

Three-dimensional ADF imaging of individual atoms by through-focal series scanning transmission electron microscopy

Aberration correction in scanning transmission electron microscopy has more than doubled the lateral resolution, greatly improving the visibility of individual impurity or dopant atoms. Depth resolution is increased five-fold, to the nanometer level. We show how a through-focal series of images enables single Hf atoms to be located inside an advanced gate dielectric device structure to a precision of better than 0.1 x 0.1 x 0.5 nm. This depth sectioning method for three-dimensional characterization has potential applications to many other fields, including polycrystalline materials, catalysts and biological structures.

Modelling imaging based on core-loss spectroscopy in scanning transmission electron microscopy

Recent experimental realizations of atomic column resolution core-loss spectroscopy in the scanning transmission electron microscope have increased the importance of routinely modelling core-loss images. We discuss different approaches to wave function simulation and how they may be used in conjunction with the mixed dynamic form factor model to simulate images resulting from such inelastic scattering events. It is shown that, as resolution improves and in situations where the degree of thermal scattering is high, detailed quantitative comparisons will require the thermal scattering of electrons to be adequately modelled. Indeed, for sufficiently strong thermal scattering even qualitative interpretation may be affected: we give an example where this leads to a contrast reversal. We describe two methods suited to this purpose, the frozen lattice model and the scattering factor model, and explain how they may be combined with the mixed dynamic form factor approach.

Nonlocality in imaging

We show how an effective nonlocality in imaging can lead to the sampling of a spatial region which is not significantly illuminated by an imaging probe. The nonlocality is embodied in the effective nonlocal potential describing inelastic scattering which occurs when coupled channel Schrödinger equations are reduced to a single integro-differential equation. The context in which this prediction will be illustrated is atomic resolution imaging based on energy-loss spectroscopy in scanning transmission electron microscopy.

The spatial resolution of imaging using core-loss spectroscopy in the scanning transmission electron microscope

The 'delocalization' of inelastic scattering is an important issue for the ultimate spatial resolution of core-loss spectroscopy in the electron microscope. This paper investigates the resolution of scanning transmission electron microscopy images for single, isolated atoms. Images are simulated from first principles using a nonlocal model for electron core-loss spectroscopy. The role of the width of the probe relative to the delocalization of the underlying ionization interaction is considered.

Limitations to the measurement of oxygen concentrations by HRTEM imposed by surface roughness

In an article published inMicroscopy and Microanalysisrecently (Jia et al., 2004), it was claimed that aberration-corrected high resolution transmission electron microscopy (HRTEM) allows the quantitative measurement of oxygen concentrations in ceramic materials with atomic resolution. Similar claims have recently appeared elsewhere, based on images obtained through aberration correction (Jia et al., 2003; Jia & Urban, 2004) or very high voltages (Zhang et al., 2003). Seeing oxygen columns is a significant achievement of great importance (Spence, 2003) that will doubtlessly allow some exciting new science; however, other models could provide a better explanation for some of the experimental data than variations in the oxygen concentration. Quantification of the oxygen concentrations was attempted by comparing experimental images with simulations in which the fractional occupancy in individual oxygen columns was reduced. The results were interpreted as representing nonstoichiometry within the bulk and at grain boundaries. This is plausible because previous studies have shown that grain boundaries can be nonstoichiometric (Kim et al., 2001), and it is indeed possible that oxygen vacancies are present at boundaries or in the bulk.However, is this the only possible interpretation?We show that for the thicknesses considered a better match to the images is obtained using a simple model of surface damage in which atoms are removed from the surface, which would usually be interpreted as surface damage or local thickness variation (from ion milling, for example).

Investigation of the effects of partial coherence on exit wave reconstruction

In a recent work we presented an iterative wave function reconstruction (IWFR) method that reconstructs a wave function from measurements of its amplitude taken as it propagates in free space (a focal series of images). Although the ideal environment for application of the IWFR method is in a coherent imaging system, it has been developed so that it can be applied in a partially coherent imaging system, in particular for a high-resolution transmission electron microscope using a field-emission gun. In this paper we investigate the effects of partial coherence on the accuracy of results obtained using the IWFR method. We then show how results obtained under such conditions can be improved by estimating and subtracting components from the amplitude measurements of the wave function that derive from incoherence in the electron beam.

Exit wave reconstruction at atomic resolution

An iterative method for exit wave function reconstruction based on wave function propagation in free space is presented. The method, which has the potential for application to many forms of microscopy, has been tailored to work with a through focal series of images measured in a high-resolution transmission electron microscope. Practical difficulties for exit wave reconstruction which are pertinent in this experimental environment are the slight incoherence of the electron beam, sample drift and its effect upon the defocus step size that can be utilised, and the number of image measurements that need to be made. To gauge the effectiveness of the method it is applied to experimental data that has been analysed previously using a maximum likelihood formalism (the MAL method).

Spectroscopic imaging of single atoms within a bulk solid

The ability to localize, identify, and measure the electronic environment of individual atoms will provide fundamental insights into many issues in materials science, physics, and nanotechnology. We demonstrate, using an aberration-corrected scanning transmission electron microscope, the spectroscopic imaging of single La atoms inside CaTiO3. Dynamical simulations confirm that the spectroscopic information is spatially confined around the scattering atom. Furthermore, we show how the depth of the atom within the crystal may be estimated.

Atomic-resolution electron energy loss spectroscopy imaging in aberration corrected scanning transmission electron microscopy

The "delocalization" of inelastic scattering is an important issue for the ultimate spatial resolution of innershell spectroscopy in the electron microscope. It is demonstrated in a nonlocal model for electron energy loss spectroscopy (EELS) that delocalization of scanning transmission electron microscopy (STEM) images for single, isolated atoms is primarily determined by the width of the probe, even for light atoms. We present experimental data and theoretical simulations for Ti L-shell EELS in a [100] SrTiO3 crystal showing that, in this case, delocalization is not significantly increased by dynamical propagation. Issues relating to the use of aberration correctors in the STEM geometry are discussed.

Channelling effects in atomic resolution STEM

A Bloch wave theory for incoherent scattering of an incident plane wave has proved successful in predicting the fine detail in 2-D zone axis channelling patterns formed by ADF, BSE and characteristic X-ray detection in beam rocking mode. A previously published example of polarity determination of GaAs by channelling contrast is compared with simulations in order to illustrate the applicability of the theory. Modification of boundary conditions for a focused coherent probe allows lattice-resolution incoherent contrast based on ADF and EELS detection as well as X-ray emissions to be catered for within a similar theoretical framework. Mixed dynamic form factors constitute an integral part of this theory, where quantum-mechanical phase is a core issue. Simulations of lattice-resolution ADF and EELS are discussed with reference to various zone axis projections of GaAs. Issues of single versus double channelling conditions, and local versus nonlocal interactions, are discussed in relation to X-ray, ADF and EELS detection.

Lattice-resolution contrast from a focused coherent electron probe. Part II

In the previous paper, boundary conditions matching the probe to the crystal wave function in scanning transmission electron microscopy were applied by matching the whole wave function across the boundary. It is shown here how that approach relates to previous Bloch wave formulations using (phase-linked) plane wave boundary conditions for wave vectors implied by the range of transverse momentum components in the incident probe. Matching the whole wave function across the boundary, and including a suitably fine mesh in the reciprocal space associated with the crystal to allow matching of transverse momentum components within the probe, leads to a structure matrix A containing many elements which would normally be excluded for plane wave incidence. For perfect crystals, the A-matrix may be block diagonalised. This leads to a considerable increase in the computational efficiency of the model and yields important insights into the physics of convergent probes in perfect crystals-reciprocity in coherent imaging and the small aperture limit for coherent and incoherent contrast are considered. The numerical equivalence of the incoherent lattice contrast calculated in this Bloch wave method and the multislice method using mixed dynamic form factors will be demonstrated. Comparison between both these methods and the frozen phonon model, a prevalent multislice method for annular dark field simulation which has the theoretical advantage of handling double channelling, will be made.

Lattice-resolution contrast from a focused coherent electron probe. Part I

To develop a Bloch wave framework for lattice-resolution contrast derived from coherent or incoherent scattering of an electron probe focused onto a crystal, boundary conditions which influence the propagation of an arbitrarily distorted coherent electron probe are addressed. These boundary conditions are particularly relevant for a probe focused within a unit cell, and lead to a general theory which hinges on Bloch wave excitation amplitudes being written as a function of beam position and focus. Whereas antisymmetric Bloch states are not excited for an incident plane wave at an exact zone axis orientation, these states may be strongly excited depending on probe focus and position within the unit cell. Equations for both coherent and incoherent lattice image contrast in scanning transmission electron microscopy are derived for any detector configuration in the Bloch wave framework. An equivalent expression amenable to evaluation via multislice techniques is also described. It is shown explicitly how mixed dynamic form factors for incoherent scattering should be taken into account for annular dark field or backscattered electron detectors, as well as for characteristic losses detected by X-ray emissions or by electron energy loss spectroscopy. A background contribution from "absorbed" electrons is included in the theory. The contribution of cross-talk from neighbouring columns to incoherent contrast is examined within the context of this theoretical framework.

Atomic scattering factors for K-shell electron energy-loss spectroscopy

Atomic scattering factors for K-shell electron energy-loss spectroscopy (EELS) have been calculated for elements in the range Z = 6 (carbon) to Z = 50 (tin). The calculations are based on relativistic Hartree-Fock wave functions for the atomic bound states and Hartree-Slater wave functions for the continuum states. The results are presented in parameterized form so that accurate values of the scattering factors can be obtained for incident electron energies between 50 and 400 keV, collection semi-angles between 10 and 40 mrad, and energy windows between 25 and 100 eV. The parameterizations are for scattering vectors with magnitude s = sin theta/lambda up to 2.5 A(-1) (2 theta is the scattering angle and lambda is the wavelength of the incident electrons).

Computational aberration correction for an arbitrary linear imaging system

We show that aberration corrections can be made in any arbitrary linear imaging system provided the aberrations are well characterized and at least one of these aberrations can be independently varied in a well-controlled manner. We derive a generalization of the Schrödinger equation for wave propagation in aberration space assuming forward scattering. Transport equations in aberration space are derived. A general iterative algorithm which can retrieve the phase, and is robust in the presence of noise, is also derived. This is demonstrated using simulated data pertinent to electron microscopy, from a series of images with differing spherical aberration.

Structure determination at the atomic level from dynamical electron diffraction data under systematic row conditions

We discuss a method to obtain structural information on crystals at the atomic level in high-resolution transmission electron microscopy from dynamical diffraction data under systematic row conditions. Working at a fixed incident energy and within an N-beam approximation, data is required at a well defined set of N incident beam orientations to determine the scattering matrix, one orientation for each column in the matrix. At each orientation the corresponding column of the scattering-matrix is obtained by Fourier transformation of the exit surface wave function. Thus, in addition to each exit surface image, we must recover the phase of the wave function for that orientation in the image plane. We show that retrieval of the phase using algorithms based on conservation of flux, which assume continuity of the phase, can yield incorrect solutions for the phase. This is because singularities can occur in the phase of the wave field at points where the intensity is zero, which can lead to edge dislocations in the phase. We demonstrate, using a model example, how these edge dislocations arise. We will show that phase retrieval from a through focal series of measurements or using the Gerchberg-Saxton algorithm (starting from measurements of an image and the corresponding diffraction pattern), correctly retrieves the phase and hence the exit surface wave function for all the orientations required to obtain the scattering-matrix. The dynamical (multiple) scattering can then be inverted to uniquely obtain the projected potential.

Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy

We discuss phase retrieval and the correction of images for aberrations, in particular defocus and spherical aberration, in high-resolution transmission electron microscopy. Non-interferometric phase retrieval requires at least two intensity measurements in different planes. Vortices in the phase may occur in the image plane or the other planes involved in the phase retrieval. We discuss the performance of various methods of phase retrieval in that case. After retrieval of the phase, the aberrations can be corrected in the Fraunhofer diffraction space (the wave function in the diffraction space is related to that in the image space by a Fourier transform). The aberration-corrected image is obtained from the aberration-corrected wave function in the diffraction plane by inverse Fourier transformation.

Inversion of dynamical electron scattering to obtain the crystal potential using data from two thicknesses

A method is given to invert the multiple scattering of electrons in a crystalline slab to obtain the projected potential. For a fixed orientation of the incident beam of electrons, scattering data are required for two thicknesses of the crystal.

Phase retrieval from images in the presence of first-order vortices

We discuss retrieval of the phase of quantum-mechanical and classical wave fields in the presence of first-order vortices. A practical method of phase retrieval is demonstrated which is robust in the presence of noise. Conditions for the uniqueness of the retrieved phase are discussed and we show that determination of the phase in a given plane requires a series of at least three two-dimensional intensity images at different propagation distances. The method is applicable to a wide range of scenarios such as the imaging of imperfect crystals, quantitative determination of the strength of vortex filaments in high-temperature superconductors, and x-ray and electron holography.