Volume 46, Issue 9, September 2005
 SPECIAL ISSUE: SUPERCONDUCTIVITY AND THE GINZBURGLANDAU MODEL

 Guest Editors: Jacob Rubinstein and Peter Sternberg


Pinning effects and their breakdown for a Ginzburg–Landau model with normal inclusions
View Description Hide DescriptionWe study a Ginzburg–Landau model for an inhomogeneous superconductor in the singular limit as the Ginzburg–Landau parameter . The inhomogeneity is represented by a potential term , with a given smooth function which is assumed to become negative in finitely many smooth subdomains, the “normally included” regions. For bounded applied fields (independent of the Ginzburg–Landau parameter ) we show that the normal regions act as “giant vortices,” acquiring large vorticity for large (fixed) applied field . For we show that this pinning effect eventually breaks down, and free vortices begin to appear in the superconducting region where , at a point set which is determined by solving an elliptic boundaryvalue problem. The associated operators are strictly but not uniformly elliptic, leading to some regularity questions to be resolved near the boundaries of the normal regions.

Emergence of the triangular lattice near a flat wall
View Description Hide DescriptionThe bifurcation of periodic solutions near a flat wall for applied magnetic fields which are slightly weaker than is considered for a reduced Ginzburg–Landau model obtained in the large limit. We formally demonstrate that following the bifurcation of the first mode, when the applied magnetic field is further decreased, there is a second bifurcation, after which the solution develops continuously into the wellknown triangular lattice.

Classical solutions to the timedependent Ginzburg–Landau equations for a bounded superconducting body in a vacuum
View Description Hide DescriptionThe initial value problem for the time dependent Ginzburg–Landau equations used to model the electrodynamics of a superconducting body surrounded by a vacuum in is studied. We prove existence, uniqueness, and regularity results for solutions in the Coulomb, Lorentz, and temporal gauges.

Surface barrier for flux entry and exit in mesoscopic superconducting systems
View Description Hide DescriptionThe energy barrier which has to be overcome for a single vortex to enter or exit the sample is studied for thin superconducting disks, rings, and squares using the nonlinear Ginzburg–Landau theory. The shape and the height of the nucleation barrier is investigated for different sample radii and thicknesses and for different values of the Ginzburg–Landau parameter . It is shown that the London theory considerably overestimates (underestimates) the energy barrier for vortex expulsion (penetration).

Timedependent Ginzburg–Landau equations with charged boundaries
View Description Hide DescriptionWe build a discrete form for the Ginzburg–Landau thermodynamic potential for an infinite film and find how the TDGL equations are related to it. We discuss why the usual superconductorinsulator boundary condition, which prohibits the passage of superconducting current, should be questioned in dynamic problems; nevertheless, we conclude that this condition remains valid. The formalism we develop enables us to deal with situations in which surface charge is present at the boundaries. These situations include the Hall configuration when the influence of the normal electrons is not negligible and the case of an electromagnetic wave parallel to the film. In the case of the electromagnetic wave, we evaluate the electromagnetic field inside the superconductor and follow the motion of the vortices.

Ginzburg–Landau model of a liquid crystal with random inclusions
View Description Hide DescriptionWe consider a Ginzburg–Landau threedimensional functional with a surface energy term to model a nematic liquid crystal with inclusions. The locations and radii of the inclusions are randomly distributed and described by a set of finite dimensional distribution functions. We show that the presence of inclusions can be accounted for by an effective potential. Our main objectives are (a) to derive the sufficient conditions on the distribution functions such that the solutions converge in probability to a solution of a homogenized deterministic problem and (b) to compute the effective potential.

Ginzburg–Landau description of confinement and quantization effects in mesoscopic superconductors
View Description Hide DescriptionAn approach to the Ginzburg–Landau problem for superconducting regular polygons is developed making use of an analytical gauge transformation for the vector potential which gives for the normal component along the boundary line of different symmetric polygons. As a result the corresponding linearized Ginzburg–Landau equation reduces to an eigenvalue problem in the basis set of functions obeying Neumann boundary condition. Such basis sets are found analytically for several symmetric structures. The proposed approach allows for accurate calculations of the order parameter distributions at low calculational cost (small basis sets) for moderate applied magnetic fields. This is illustrated by considering the nucleation of superconductivity in squares, equilateral triangles and rectangles, where vortex patterns containing antivortices are obtained on the phase boundary. The calculated phase boundaries are compared with the experimental curves measured for squares, triangles, disks, rectangles, and loops. The stability of the symmetry consistent solutions against small deviations from the phase boundary line deep into the superconducting state is investigated by considering the full Ginzburg–Landau functional. It is shown that below the nucleation temperature symmetryswitching or symmetrybreaking phase transitions can take place. The symmetrybreaking phase transition has the same structure as the pseudoJahnTeller instability of high symmetry nuclear configurations in molecules. The existence of these transitions is predicted to be strongly dependent on the size of the samples.

Numerical approximations of the Ginzburg–Landau models for superconductivity
View Description Hide DescriptionIn this paper, we review various methods for the numerical approximations of the Ginzburg–Landau models of superconductivity. Particular attention is given to the different treatment of gauge invariance in both the finite element, finite difference, and finite volume settings. Representative theoretical results, typical numerical simulations, and computational challenges are presented. Generalizations to other relevant models are also discussed.

Topologically driven local minimizers of the OseenFrank energy
View Description Hide DescriptionThe analogy between certain liquid crystals and superconductivity has been recognized and explored by a number of scientists. In particular, mathematical techniques first developed within the Ginzburg–Landau theory of superconductivity have proven useful when adapted to the setting of liquid crystals. Here we pursue nontrivial stable liquid crystal configurations, motivated by an approach used by the authors in the setting of Ginzburg–Landau to produce persistent currents in topologically nontrivial domains. Our starting point is the OseenFrank energy for a nematic, but we add to the standard model a term that penalizes deviation of the director from a given plane.

Global bifurcation structure of a onedimensional Ginzburg–Landau model
View Description Hide DescriptionWe consider an equation of a simplified Ginzburg–Landau model of superconductivity in a onedimensional ring. The equation for a complex order parameter has two real parameters and related to the magnitude of an applied magnetic field and the Ginzburg–Landau parameter, respectively. The purpose of this paper is to reveal a global bifurcation structure for the equation in the parameter space . In particular we show that there exist modulating amplitude solutions which bifurcate from constant amplitude solutions, and how the bifurcation branches of such solutions continue or disappear as varies. We also determine the minimizer of the energy functional.

Bifurcation and stability of superconductivity^{a)}
View Description Hide DescriptionIn this article, we present a bifurcation and stability analysis on timedependent Ginzburg–Landau model of superconductivity. It is proved in particular that there are two different phase transitions from the normal state to superconducting states or vice versa: one is continuous, and the other is jump. These two transitions are precisely determined by a simple nondimensional parameter, which links the superconducting behavior with the geometry of the material, the applied field and the physical parameters. The rigorous analysis is conducted using a bifurcation theory newly developed by the authors, and provides some interesting physical predictions.