Atomic electron tomography (AET) has become a powerful tool for atomic-scale structural characterization in three and four dimensions. It provides the ability to correlate structures and properties of materials at the single-atom level. With recent advances in data acquisition methods, iterative three-dimensional (3D) reconstruction algorithms, and post-processing methods, AET can now determine 3D atomic coordinates and chemical species with sub-Angstrom precision, and reveal their atomic-scale time evolution during dynamical processes. Here, we review the recent experimental and algorithmic developments of AET and highlight several groundbreaking experiments, which include pinpointing the 3D atom positions and chemical order/disorder in technologically relevant materials and capturing how atoms rearrange during early nucleation at four-dimensional atomic resolution.

Electron tomography offers useful three-dimensional (3D) structural information, which cannot be observed by two-dimensional imaging. By combining annular dark-field scanning transmission electron microscopy (ADF STEM) with aberration correction, the resolution of electron tomography has reached atomic resolution. However, tomography based on ADF STEM inherently suffers from several issues, including a high electron-dose requirement, poor contrast for light elements, and artifacts from image-contrast nonlinearity. Here, we develop an alternative method called multislice electron tomography (MSET) based on four-dimensional STEM tilt series. Our simulations show that multislice-based 3D reconstruction can effectively reduce undesirable reconstruction artifacts from the nonlinear contrast, allowing precise determination of atomic structures with improved sensitivity for low-Z elements, at considerably low electron-dose conditions. We expect that the MSET method can be applied to a wide variety of materials, including radiation-sensitive samples and materials containing light elements whose 3D atomic structures have never been fully elucidated due to electron-dose limitations or nonlinear imaging contrast.

In the early 2000s, low dimensional ferroelectric systems were predicted to have topologically nontrivial polar structures, such as vortices or skyrmions, depending on mechanical or electrical boundary conditions. A few variants of these structures have been experimentally observed in thin film model systems, where they are engineered by balancing electrostatic charge and elastic distortion energies. However, the measurement and classification of topological textures for general ferroelectric nanostructures have remained elusive, as it requires mapping the local polarization at the atomic scale in three dimensions. Here we unveil topological polar structures in ferroelectric BaTiO3 nanoparticles via atomic electron tomography, which enables us to reconstruct the full three-dimensional arrangement of cation atoms at an individual atom level. Our three-dimensional polarization maps reveal clear topological orderings, along with evidence of size-dependent topological transitions from a single vortex structure to multiple vortices, consistent with theoretical predictions. The discovery of the predicted topological polar ordering in nanoscale ferroelectrics, independent of epitaxial strain, widens the research perspective and offers potential for practical applications utilizing contact-free switchable toroidal moments.

Tomography has made a radical impact on diverse fields ranging from the study of 3D atomic arrangements in matter to the study of human health in medicine. Despite its very diverse applications, the core of tomography remains the same, that is, a mathematical method must be implemented to reconstruct the 3D structure of an object from a number of 2D projections. Here, we present the mathematical implementation of a tomographic algorithm, termed GENeralized Fourier Iterative REconstruction (GENFIRE), for high-resolution 3D reconstruction from a limited number of 2D projections. GENFIRE first assembles a 3D Fourier grid with oversampling and then iterates between real and reciprocal space to search for a global solution that is concurrently consistent with the measured data and general physical constraints. The algorithm requires minimal human intervention and also incorporates angular refinement to reduce the tilt angle error. We demonstrate that GENFIRE can produce superior results relative to several other popular tomographic reconstruction techniques through numerical simulations and by experimentally reconstructing the 3D structure of a porous material and a frozen-hydrated marine cyanobacterium. Equipped with a graphical user interface, GENFIRE is freely available from our website and is expected to find broad applications across different disciplines.

Crystallography, the primary method for determining the 3D atomic positions in crystals, has been fundamental to the development of many fields of science. However, the atomic positions obtained from crystallography represent a global average of many unit cells in a crystal. Here, we report, for the first time, the determination of the 3D coordinates of thousands of individual atoms and a point defect in a material by electron tomography with a precision of ∼19 pm, where the crystallinity of the material is not assumed. From the coordinates of these individual atoms, we measure the atomic displacement field and the full strain tensor with a 3D resolution of ∼1 nm(3) and a precision of ∼10(-3), which are further verified by density functional theory calculations and molecular dynamics simulations. The ability to precisely localize the 3D coordinates of individual atoms in materials without assuming crystallinity is expected to find important applications in materials science, nanoscience, physics, chemistry and biology.

Nanomaterials with core-shell architectures are prominent examples of strain-engineered materials. The lattice mismatch between the core and shell materials can cause strong interface strain, which affects the surface structures. Therefore, surface functional properties such as catalytic activities can be designed by fine-tuning the misfit strain at the interface. To precisely control the core-shell effect, it is essential to understand how the surface and interface strains are related at the atomic scale. Here, we elucidate the surface-interface strain relations by determining the full 3D atomic structure of Pd@Pt core-shell nanoparticles at the single-atom level via atomic electron tomography. Full 3D displacement fields and strain profiles of core-shell nanoparticles were obtained, which revealed a direct correlation between the surface and interface strain. The strain distributions show a strong shape-dependent anisotropy, whose nature was further corroborated by molecular statics simulations. From the observed surface strains, the surface oxygen reduction reaction activities were predicted. These findings give a deep understanding of structure-property relationships in strain-engineerable core-shell systems, which can lead to direct control over the resulting catalytic properties.

Perfect crystals are rare in nature. Real materials often contain crystal defects and chemical order/disorder such as grain boundaries, dislocations, interfaces, surface reconstructions and point defects. Such disruption in periodicity strongly affects material properties and functionality. Despite rapid development of quantitative material characterization methods, correlating three-dimensional (3D) atomic arrangements of chemical order/disorder and crystal defects with material properties remains a challenge. On a parallel front, quantum mechanics calculations such as density functional theory (DFT) have progressed from the modelling of ideal bulk systems to modelling 'real' materials with dopants, dislocations, grain boundaries and interfaces; but these calculations rely heavily on average atomic models extracted from crystallography. To improve the predictive power of first-principles calculations, there is a pressing need to use atomic coordinates of real systems beyond average crystallographic measurements. Here we determine the 3D coordinates of 6,569 iron and 16,627 platinum atoms in an iron-platinum nanoparticle, and correlate chemical order/disorder and crystal defects with material properties at the single-atom level. We identify rich structural variety with unprecedented 3D detail including atomic composition, grain boundaries, anti-phase boundaries, anti-site point defects and swap defects. We show that the experimentally measured coordinates and chemical species with 22 picometre precision can be used as direct input for DFT calculations of material properties such as atomic spin and orbital magnetic moments and local magnetocrystalline anisotropy. This work combines 3D atomic structure determination of crystal defects with DFT calculations, which is expected to advance our understanding of structure-property relationships at the fundamental level.

Nucleation plays a critical role in many physical and biological phenomena ranging from crystallization, melting and evaporation to the formation of clouds and the initiation of neurodegenerative diseases. However, nucleation is a challenging process to study in experiments especially in the early stage when several atoms/molecules start to form a new phase from its parent phase. Here, we advance atomic electron tomography to study early stage nucleation at 4D atomic resolution. Using FePt nanoparticles as a model system, we reveal that early stage nuclei are irregularly shaped, each has a core of one to few atoms with the maximum order parameter, and the order parameter gradient points from the core to the boundary of the nucleus. We capture the structure and dynamics of the same nuclei undergoing growth, fluctuation, dissolution, merging and/or division, which are regulated by the order parameter distribution and its gradient. These experimental observations differ from classical nucleation theory (CNT) and to explain them we propose the order parameter gradient (OPG) model. We show the OPG model generalizes CNT and energetically favours diffuse interfaces for small nuclei and sharp interfaces for large nuclei. We further corroborate this model using molecular dynamics simulations of heterogeneous and homogeneous nucleation in liquid-solid phase transitions of Pt. We anticipate that the OPG model is applicable to different nucleation processes and our experimental method opens the door to study the structure and dynamics of materials with 4D atomic resolution.

Nucleation plays a critical role in many physical and biological phenomena that range from crystallization, melting and evaporation to the formation of clouds and the initiation of neurodegenerative diseases^1-3. However, nucleation is a challenging process to study experimentally, especially in its early stages, when several atoms or molecules start to form a new phase from a parent phase. A number of experimental and computational methods have been used to investigate nucleation processes^4-17, but experimental determination of the three-dimensional atomic structure and the dynamics of early-stage nuclei has been unachievable. Here we use atomic electron tomography to study early-stage nucleation in four dimensions (that is, including time) at atomic resolution. Using FePt nanoparticles as a model system, we find that early-stage nuclei are irregularly shaped, each has a core of one to a few atoms with the maximum order parameter, and the order parameter gradient points from the core to the boundary of the nucleus. We capture the structure and dynamics of the same nuclei undergoing growth, fluctuation, dissolution, merging and/or division, which are regulated by the order parameter distribution and its gradient. These experimental observations are corroborated by molecular dynamics simulations of heterogeneous and homogeneous nucleation in liquid-solid phase transitions of Pt. Our experimental and molecular dynamics results indicate that a theory beyond classical nucleation theory^1,2,18 is needed to describe early-stage nucleation at the atomic scale. We anticipate that the reported approach will open the door to the study of many fundamental problems in materials science, nanoscience, condensed matter physics and chemistry, such as phase transition, atomic diffusion, grain boundary dynamics, interface motion, defect dynamics and surface reconstruction with four-dimensional atomic resolution.

Heterogeneous nanocatalysts play a crucial role in both the chemical and energy industries. Despite substantial advancements in theoretical, computational and experimental studies, identifying their active sites remains a major challenge. Here we utilize atomic electron tomography to determine the three-dimensional atomic structure of PtNi and Mo-doped PtNi nanocatalysts for the electrochemical oxygen reduction reaction. We then employ the experimental atomic structures as input to first-principles-trained machine learning to identify the active sites of the nanocatalysts. Through the analysis of the structure–activity relationships, we formulate an equation termed the local environment descriptor, which balances the strain and ligand effects to provide physical and chemical insights into active sites in the oxygen reduction reaction. The ability to determine the three-dimensional atomic structure and chemical composition of realistic nanoparticles, combined with machine learning, could transform our fundamental understanding of the active sites of catalysts and guide the rational design of optimal nanocatalysts. (Figure presented.)