Characterizing Self-assembled Nanostructures via Shapelet Functions

Abstract

Pattern formation is a natural phenomena that occurs at various length-scales. Lattice patterns are a particular type composed of spatially-repeating features with stripe, square, or hexagonal symmetries. They are of particular interest to nanotechnology researchers due to their frequent appearance in self-assembly and lithography processes. Self-assembled nanostructures provide many technological applications but are difficult to characterize due to deformations in local structure (defects, disorder). While image-based characterization techniques for nanostructures are well-known (i.e., scanning electron microscopy), appropriate computational techniques to characterize their structure are seldom developed and are typically without readily available open-source implementations.

Characterization of self-assembled nanostructures is important to develop structure-property relationships with potential to advance defect engineering research. Defect engineering corresponds to the regulation of specific defects within nanostructure to manipulate material properties (physical, chemical, magnetic) and improve material functionality.

Existing techniques to characterize self-assembled nanostructures, including Voronoi diagrams/entropy, bond-orientational order theory, and Fourier space filtering are well-known but contain inherent limitations. A more recent and promising approach uses a set of localized basis functions called shapelets, originally designed for the compression and reconstruction of images of galaxies. This approach uses polar shapelets, providing unique rotational/radial symmetry properties beneficial for analysis on non-Euclidean geometries. This response distance method is a supervised learning technique that quantifies local deformations in structure (defects, disorder) apart from regions displaying uniform pattern order.

This work presents extensions to the existing response distance method, including a decrease in computational runtime, along with the inclusion of higher-order shapelet functions to improve order quantification in areas with topological defects. New shapelet-based methods are also presented, such as quantification of local pattern orientation and a technique to directly identify topological defects and defect structures. These methods are validated against both simulated and experimental surface images of self-assembled nanostructures containing stripe, square, and hexagonal patterns and demonstrates their effectiveness in the presence of measurement noise. Furthermore, they are made available to the community as part of an open-source Python library, along with reference implementation of other shapelet functions and applications to promote collaboration and transparency in shapelet research. The mathematical framework for a higher-order shapelet class with radial symmetry is also provided, laying the foundation for future pattern analysis.