Identification of Three-Dimensional Crystal Lattices by Estimation of Their Unit Cell Parameters

Abstract. The problem of the identification of three-dimensional crystal lattices is considered in the article. Two matching methods based on estimation of unit cell parameters were developed to solve this problem. The first method estimates and compares main parameters of Bravais unit cells. The second method estimates and compares volumes of Wigner-Seitz unit cells. Both methods include normalised similarity measures: an edge similarity measure and an angle similarity measure for Bravais cells and a volume similarity measure for Wigner-Seitz cells. The results of computational experiments on the large set of simulated lattices showed that the developed methods allowed to achieve the identification accuracy above 90% for four lattice systems

Atom-Based Geometrical Fingerprinting of Conformal Two-Dimensional Materials

The shape of two-dimensional materials plays a significant role on their chemical and physical properties. Two-dimensional materials are basic meshes that are formed by mesh points (vertices) given by atomic positions, and connecting lines (edges) between points given by chemical bonds. Therefore the study of local shape and geometry of two-dimensional materials is a fundamental prerequisite to investigate physical and chemical properties. Hereby the use of discrete geometry to discuss the shape of two-dimensional materials is initiated. The local geometry of a surface embodied in 3D space is determined using four invariant numbers from the metric and curvature tensors which indicates how much the surface is stretched and curved under a deformation as compared to a reference pre-deformed conformation. Many different disciplines advance theories on conformal two-dimensional materials by relying on continuum mechanics and fitting continuum surfaces to the shape of conformal two-dimensional materials. However two-dimensional materials are inherently discrete. The continuum models are only applicable when the size of two-dimensional materials is significantly large and the deformation is less than a few percent. In this research, the knowledge of discrete differential geometry was used to tell the local shape of conformal two-dimensional materials. Three kind of two-dimensional materials are discussed: 1) one atom thickness structures such as graphene and hexagonal boron nitride; 2) high and low buckled 2D meshes like stanene, leadene, aluminum phosphate; and, 3) multi layer 2D materials such as Bi2Se3 and WSe2. The lattice structures of these materials were created by designing a mechanical model - the mechanical model was devised in the form of a Gaussian bump and density-functional theory was used to inform the local height; and, the local geometries are also discussed

Crystal image analysis using 2D2D synchrosqueezed transforms

We propose efficient algorithms based on a band-limited version of 2D synchrosqueezed transforms to extract mesoscopic and microscopic information from atomic crystal images. The methods analyze atomic crystal images as an assemblage of non-overlapping segments of 2D general intrinsic mode type functions, which are superpositions of non-linear wave-like components. In particular, crystal defects are interpreted as the irregularity of local energy; crystal rotations are described as the angle deviation of local wave vectors from their references; the gradient of a crystal elastic deformation can be obtained by a linear system generated by local wave vectors. Several numerical examples of synthetic and real crystal images are provided to illustrate the efficiency, robustness, and reliability of our methods.Comment: 27 pages, 17 figure

Universal classification of twisted, strained and sheared graphene moir\'e superlattices

Moir\'e superlattices in graphene supported on various substrates have opened a new avenue to engineer graphene's electronic properties. Yet, the exact crystallographic structure on which their band structure depends remains highly debated. In this scanning tunneling microscopy and density functional theory study, we have analysed graphene samples grown on multilayer graphene prepared onto SiC and on the close-packed surfaces of Re and Ir with ultra-high precision. We resolve small-angle twists and shears in graphene, and identify large unit cells comprising more than 1,000 carbon atoms and exhibiting non-trivial nanopatterns for moir\'e superlattices, which are commensurate to the graphene lattice. Finally, a general formalism applicable to any hexagonal moir\'e is presented to classify all reported structures.Comment: 14 pages, 6 figure

Multi-field approach in mechanics of structural solids

We overview the basic concepts, models, and methods related to the multi-field continuum theory of solids with complex structures. The multi-field theory is formulated for structural solids by introducing a macrocell consisting of several primitive cells and, accordingly, by increasing the number of vector fields describing the response of the body to external factors. Using this approach, we obtain several continuum models and explore their essential properties by comparison with the original structural models. Static and dynamical problems as well as the stability problems for structural solids are considered. We demonstrate that the multi-field approach gives a way to obtain families of models that generalize classical ones and are valid not only for long-, but also for short-wavelength deformations of the structural solid. Some examples of application of the multi-field theory and directions for its further development are also discussed.Comment: 25 pages, 18 figure