The breakdown of superconductivity in high magnetic fields
Abstract
The Ginzburg-Landau model is used to study the breakdown of superconductivity in materials subjected to high constant magnetic fields. It is experimentally known that if the intensity of the external field is high enough superconductivity is lost, i.e., the material is in the normal state. The behavior for high applied magnetic fields of the solutions of the semilinear elliptic Ginzburg-Landau system is considered. The analysis is carried out for the case of an infinite cylinder of superconducting material having a possibly multiply-connected cross-section with a somewhat regular boundary (for symmetry reasons, this is essentially a two-dimensional problem), and for a three-dimensional multiply-connected bounded superconductor, again with regular boundary. In Chapter 2, for the two-dimensional case the Ginzburg-Landau energy functional is studied, and a uniform lower bound used in Chapter 4 is derived. Chapter 3 deals with the case of a finite three-dimensional body. The problem here is truly three-dimensional, and the introduction of different functional spaces is required. Results similar to the ones of Chapter 1 are obtained. Chapter 4 contains an eigenvalue estimate which implies the existence of a critical field above which the only solutions to the Ginzburg-Landau equations are the normal state solutions. Chapter 5 is devoted to the search for an upper bound for the critical field described in Chapter 4. In the literature, superconductors are classified based on their different magnetic properties as type I and type II. For type II superconductors the upper bound found is linear in the Ginzburg-Landau parameter $\kappa,$ in agreement with experimental data.
Degree
Ph.D.
Advisors
Phillips, Purdue University.
Subject Area
Mathematics
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