Abstract: Superconductivity is one of the most important discoveries of the last century. With many applications in physics, engineering, and technology, superconductors are crucial to our way of living. Several material and engineering issues however prevent their widespread usage in everyday life. Comprehensive studies are being directed at these materials and their properties to come up with new technologies that will address these challenges and enhance their superconductive capabilities. In this context, numerical modeling plays an important role in the search of new solutions to existing material and engineering issues. The time-dependent Ginzburg-Landau (TDGL) theory is a powerful predictive tool for modeling the macroscopic behavior of superconductors. However most of the numerical algorithms developed so far are incapable of describing many basic properties of real superconducting devices, and are too slow on current hardware for large-scale numerical simulations necessary for their accurate description. Therefore, the purpose of this thesis is to develop high-performing numerical solutions that can correctly describe material features to be used as modeling tools of laboratory experiments. Some important innovations introduced in this work include the numerical modeling of nonrectangular geometrical shapes with complex electrical and insulating components, the inclusion of dynamic heating of the material, and the description of different types of material inhomogeneities. These encompass the principal features necessary for a complete description of the superconductive physics in real material samples. In this thesis a numerical solution is developed for modeling superconducting thin films and used to study the superconductive properties of three experimental configurations: the dynamics of vortex matter in a Corbino disk, the motion of ultrafast vortices in an hourglass-shaped microbridge, and the photon detection process in a meander-patterned nanowire. Moreover, a numerical solution is developed for modeling three-dimensional superconductors which are studied here for the first time in the type-I superconducting regime. These numerical algorithms are optimized to exploit the computational horsepower of graphics processing units (GPUs) and multicore central-processing unit (CPU) clusters such that they can achieve high-performance and be used to model large-scale problems previously impossible on conventional machines. Several computational tools are also designed to assist with the modeling of superconducting devices. These include a numerical library of the TDGL equations, a novel mechanism for the generation of complex geometries, a closed-form solver to conduct numerical simulations, and a graphics user interface (GUI) to visualize the dynamic behavior of superconductors. The contributions in this thesis ultimately push the boundaries on what is possible in state-of-the-art numerical modeling of superconductivity.

Abstract: Moore’s Law has driven the electronics industry for the past half century. However, the doubling of transistors about every two years is beginning to break down, owing to fundamental limits that arise as they approach the atomic length. As a result, the search for new pathways for electronics has become crucial. Among potential candidates, the discovery of magnetic textures known as skyrmions has attracted considerable interest and attention in spintronic technology, which relies on both the electron charge and its spin. The unusual topological and particle-like behavior launched skyrmions into the spotlight of scientific research. Topological protected stability, nanoscale size, and low driving currents needed to move them make skyrmions promising candidates for future consumer nanoelectronics. Recent advances in the field have provided all of the basic functions needed for carrying and processing information. In this thesis, we procure to advance the current understanding of skyrmion physics, and explore their potential to replace conventional electronics technology. First, the fundamental properties and lifetimes of racetrack skyrmions at room temperature are investigated. We discover that skyrmions can easily collapse at the boundary in laterally finite systems, and propose ways to improve their stability for constrained geometries. Then, pinning of single skyrmions on atomic defects of distinct origins are studied. We reveal that the preferred pinning positions depend on the skyrmion size and type of defect being considered, and discuss applications where control of skyrmions by defects is of particular interest. Next, we explore other magnetic configurations that can compete with skyrmions when considering new materials, and describe a previously unseen mechanism for collapse of skyrmions into cycloidal spin backgrounds. Finally, switching and interactions between skyrmions with distinct topologies are reported. We find that skyrmions transition to higher or lower topologies by absorbing a unit spin texture. The interactions between skyrmions of different topological charges can be attractive or repulsive, leading to the formation of arranged clusters. We conclude with a numerical library for simulating magnetic skyrmions in various scenarios.