Computational Modeling of Superconductivity from the Set of Time-Dependent Ginzburg-Landau Equations for Advancements in Theory and Applications
Date of Award
Doctor of Philosophy (PhD)
Computational and Data Sciences
Dr. Armen Gulian
Dr. Domenico Napoletani
Dr. Thomas Piechota
A full review of the research conducted and published during my PhD studies in Computational and Data Sciences at Chapman University, under the advisement of Dr. Armen Gulian, are presented. Using the set of time-dependent Ginzburg-Landau (TDGL) equations with inclusion of the interference current and the non-equilibrium phonon term, we modeled the dynamics of superconductors in various theory revealing states and practical purposes. A review of the history and phenomenon of superconductivity, including modern applications, is introduced. The Josephson effect and associated Josephson junction are discussed for comparison to our analogous results with the 1-D superconducting wire. The mathematics of microscopic theory of superconductivity are explored with mention of BCS theory. The time-dependent Ginzburg-Landau theory is detailed, as well as an interpretation from Dr. Aharonov’s discussion with my advisor on how the simplest TDGL equation is equivalent to the Schrödinger equation with imaginary Hamiltonian. Methods of how to transform the TDGL equations into dimensionless form and implement into the modeling software COMSOL Multiphysics® for a 1-D wire model are revealed in tutorial form. I present my research in the order of increased dimensions: 1-D, 2-D, and 3-D models, with details of solving each higher dimension with different gauge choices. The results for the 1-D model are the visualization of the 1-D wire dynamics of phase slip centers (PSCs), the pitchfork bifurcation of PSC locations at varying energy gaps, the influence of phonon flux to the PSC, and the proposed application of a 1-D wire cryocooler from these previous results. The published result of the superconducting diode effect shows vortices entering a 2-D slab unidirectional and aided in explaining the microscopic theory behind the lab results at the Advanced Physics Laboratory. The 3-D published results were for the use of stacks of superconducting rings as transducers for the gravitational wave detector, GEFEST, and how best to situate them for highest performance of the detector. The other 3-D model showed how superconducting nanorings, due to their size and closest to the critical temperature, violate conservation of magnetic flux when of dimensional size of the order of λL.
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I. Mowgood, "Computational modeling of superconductivity from the set of time-dependent Ginzburg-Landau equations for advancements in theory and applications," Ph.D. dissertation, Chapman University, Orange, CA, 2023. https://doi.org/10.36837/chapman.000483