Vortices in the Ginzburg-Landau Superconductivity Model

Open Access
- Author:
- Iaroshenko, Oleksandr
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- February 28, 2018
- Committee Members:
- Leonid V Berlyand, Dissertation Advisor/Co-Advisor
Leonid V Berlyand, Committee Chair/Co-Chair
Anna L Mazzucato, Committee Member
Helge Kristian Jenssen, Committee Member
Kenneth O'Hara, Outside Member - Keywords:
- superconductivity
Ginzburg-Landau
variational problems
vortices
pinning - Abstract:
- In this dissertation we analyze the behavior of vortices in superconductors. The vortices might appear in a superconductor when it is immersed in a magnetic field and lead to the energy dissipation which makes them important to study. In Chapter 1 we discuss the main concepts related to superconductivity and introduce the Ginzburg-Landau model that describes this phenomenon. We explain the nature of superconductivity and the Ginzburg-Landau model in both physical and mathematical aspects. The model itself describes the ground state of the superconductor that is the lowest energy state. Mathematically the ground state can be found via a minimization problem, where an energy functional is being minimized among all possible physical states to achieve the lowest possible value. The main focus of this work are the vortices that appear in superconductor as certain singularities when the applied magnetic field is strong enough. The appearance of vortices leads to the energy dissipation, therefore we introduce a way to suppress it via the pinning effect. Pinning happens when there is a columnar defect (CD) in a superconductor that traps nearby vortices to itself. In this work we consider a superconductor of a cylindrical shape so that it can be modeled by a 2D cross section. The columnar defects are then the cylindrical tunnels of damaged material, material with different conductivity, or an empty holes, that extend through the sample. Superconductivity is weakened in CDs and the energy cost of locating a vortex in this region is reduced. In Chapter 2 (with L. Berlyand, V. Rybalko, and V. Vinokur [1]) we show that a particular configuration of the geometry of a superconducting sample as well as the strength of the magnetic field leads to the pinning effect which presents an unexpected distribution of pinned vortices. The CDs are arranged in a periodic lattice with a small period and mathematically modeled as holes with even smaller radii. We call the vortices in the CDs hole vortices as opposed to bulk vortices that appear outside of them. The degree of the hole vortex is a mathematical concept that essentially counts the number of vortices trapped in a CD. We assume that all vortices are pinned and none of them appear outside of CDs. This is iii modeled using a constraint |u| = 1, where u is the order parameter, the function that describes two conducting phases in a sample. Essentially this means that there is a perfect superconducting phase outside of CDs. Under this assumption, we find that a superconductor admits a hierarchical nested domain structure where these domains have a different average number of vortices pinned at each CD inside them. The average number of vortices follows the integer sequence starting at 0 in the outer subdomain with the only exception for the most inner subdomain where this number might be fractional. This ground state is completely different from what was observed before both experimentally and analytically. Chapter 3 (with D. Golovaty, V. Rybalko, and L. Berlyand [2]) justifies the assumption used in Chapter 2. We consider a similar setting to the one used in Chapter 2 with CDs arranged randomly instead of periodically. The number of CDs is fixed and does not increase when their size becomes smaller. We show that this setting leads to the absence of vortices outside of CDs. Moreover, the absence of bulk vortices and the potential term in the Ginzburg-Landau energy suggest that the absolute value of the order parameter u should be close to 1 at the majority of the domain. This is shown by considering a constraint |u| = 1 on the Ginzburg-Landau energy minimization problem. We prove that the energies of both unconstrained and constrained problems are close to each other and their minimizers have the same degrees of the hole vortices at the corresponding CDs. This result allows us to consider a simpler S1-valued problem when we need to find the distribution of the degrees of the vortices that is exactly what was done in order to find the nested structure of subdomains in Chapter 2.