Nanoscale Patterns: Topology, Geometry, and Physical Mechanisms
When a solid surface is bombarded with a broad ion beam, self-assembled patterns, such as hexagonal arrays of nanodots, form. Ion bombardment could become a widely employed method of fabricating large-scale nanostructures with wavelengths as short as 10 nm if these patterns were not typically ridden with defects. We build a model for nanoscale pattern formation in binary materials. Motivated by understanding and controlling defects in these as well as other nanoscale patterns, we develop quantitative measures of order for imperfect Bravais lattices in the plane that help compare the model with experimental data. A tool from topological data analysis called persistent homology, combined with a metric on point distributions, are the key components for defining these measures. We also develop a method, called persistent images, that employs persistent homology and machine learning to help determine parameter values from experimental data. Of particular interest is understanding the role of the local geometry compared to the global topology of the data in the method of persistent images.