# Superconductivity

### Faculty

Qiang Du*, Max Gunzburger**, Janet Peterson
* Research supported in part by NSF-DMS9798206
** Research supported in part by NSF, AFSOR

### Description

We have achieved significant progress in the modeling, analysis, and computation of superconducting phenomena. Our work has focused on mezoscale models as typified by the celebrated Ginzburg-Landau equations; these models are intermediate between the microscopic models (that can be used to understand the basic structure of superconductors and of the atomic and sub-atomic behavior of these materials) and the macroscale, or homogenized, models (that can be of use for the design of devices). Mezoscale models are of great use in understanding important phenomena in superconductors such as pinning, vortex motion, critical currents, Josephson effects, fluctuations, vortex glass and vortex fluid structures, resistivity, etc.
The goals of our past, present, and future efforts in superconductivity center on the following aspects:
• to develop or refine mezoscale and macroscale models for superconductivity so to enlarge the range of physical problems for which such models are valid;
• to analyze these models in order to gain further understanding of the properties of these models and of their solutions, and also to determine their validity and usefulness for solving physically interesting problems;
• to develop, analyze, and implement algorithms for the numerical simulation of solutions of the various models;
• to use our algorithms and codes to study superconducting phenomena.
Currently, we are working closely with scientists at the Ames Laboratory; this work invovles the application of our algorithms and codes to the solution of important problems in superconductivity. We now give an outline, roughly divided by problem characteristics, of past work in superconductivity. More details may be found in the references cited below.
Time dependent Ginzburg-Landau model
New theoretical results have been obtained, and finite element algorithms have been defined, analyzed, and implemented. In addition, we have developed a code for approximate solutions of this model which forms the basis for all of our computational studies. We have, for example, incorporated the ability to handle applied voltages and currents, and other intrinsically transient superconducting phenomena.

Picture of TDGL calculations
Videos of TDGL calculations
Movie1
Movie2
Movie3

Variable thickness thin film model
A new model for thin films having variable thickness has been developed; this model simplifies our own previous model for this situation. Theoretical results have been obtained, and finite element algorithms have been defined, analyzed, and implemented. Computational simulations showing the effectiveness of thin regions in the pinning of vortices have been carried out.

Pictures of thin film calculations

Models for high values of the Ginzburg-Landau parameter
Simplified models have been developed and analyzed that are valid for large values of the Ginzburg-Landau parameter. Since high critical temperature superconductors are known to have large values of of this parameter, this model has become one of our basic tools in the study of such materials. For example, our Lawrence-Doniach code is based on a high Ginzburg-Landau parameter version of that model.

Pictures for high Ginzburg-Landau parameter calculations

Lawrence-Doniach model
The connection between this model for layered superconductors and the anisotropic Ginzburg-Landau model has been rigorously established. New theoretical results have been obtained, and finite element algorithms have been defined. Preliminary computational simulations have been carried out.

Movies for Laurence-Doniach model
Movie4
Movie5

Anisotropic Ginzburg-Landau model
Various theoretical results about this model for materials having anisotropic material properties have been derived. These include its connection to the Lawrence-Doniach model and a rigorous demonstration of certain scaling properties that relate this model the isotropic Ginaburg-Landau model. Computational simulations have been carried out.

Pictures for anisotropic Ginzburg-Landau model

Models accounting for normal inclusions
Models that can account for normal materials such as impurities or layers co-existing with superconducting materials have been developed and analyzed. Computational simulations showing the effectiveness of normal impurities in the pinning of vortices have been carried out. The same code has been used to show how the model we have developed can be used for the study of Josephson junctions and and de-pinning phenomena. Some preliminary calculations for these effects have been carried out.

Pictures for normal inclusion model
Movies for normal inclusion model
Movie6
Movie7
Movie8
Movie9
Movie10

Models accounting for thermal fluctuations
Models that can account for thermal fluctuation have been developed an are being analyzed. Fluctuations are modeled through Langevin terms in the order parameter equation. Some preliminary computational results have been obtained.

Movies for thermal fluctuation model
Movie11

Mean Field Models

$d$-wave Models

## References

Selected publications in superconductivity

Analysis and approximation of the Ginzburg-Landau model of superconductivity; SIAM Review 34 1992, 54-81; Q. Du, M. Gunzburger and J. Peterson.

Solving the Ginzburg-Landau equations by finite element methods; Phys. Rev. B 46 1992, 9027-9034; Q. Du, M. Gunzburger and J. Peterson.

Finite element approximation of a periodic Ginzburg-Landau model for type-II superconductors; Numer. Math. 64 1993, 85-114; with Q. Du, M. Gunzburger and J. Peterson.

Modeling and analysis of a periodic Ginzburg-Landau model for type-II superconductors; SIAM J. Applied Math. 53 1993, 689-717; Q. Du, M. Gunzburger and J. Peterson.

A model for superconducting thin films having variable thickness; Physica D 69 1993, 215-231; Q. Du and M. Gunzburger.

On the Lawrence-Doniach and anisotropic Ginzburg-Landau models for layered superconductors; SIAM J. Appl. Math. 55, 1995, 156-174; S. Chapman, Q. Du and M. Gunzburger.

Computational simulation of type-II superconductivity including pinning phenomena; Phys. Rev. B 51 1995, 16194-16203; Q. Du, M. Gunzburger and J. Peterson.

A Ginzburg-Landau type model of superconducting/normal junctions including Josephson junctions; Europ. J. Appl. Math. 6 1995, 97-114; S. Chapman, Q. Du and M. Gunzburger.

Simplified Ginzburg-Landau type models of superconductivity in the high kappa, high field limit; Adv. Math. Sciences Appl. 1995, 193-218; S. Chapman, Q. Du, M. Gunzburger and J. Peterson.

A model for variable thickness superconducting thin films; ZAMP 47, 1996, 410-431; S. Chapman, Q. Du and M. Gunzburger.

Analysis and approximation of optimal control problems for a simplified Ginzburg-Landau model of superconductivity; to appear in Numer. Math.; M. Gunzburger, L. Hou and S. Ravindran.

The critical temperature and gap solution of the Bardeen-Cooper-Schrieffer theory of superconductivity; Lett. Math. Phys. 29, pp133-150, 1993; Q. Du and Y. Yang.

Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau equations in superconductivity Applicable Analysis,53, 1-2, pp1-17, (1994) ; Q. Du.

High-kappa limit of the time dependent Ginzburg-Landau model for superconductivity, SIAM J. Appl. Math. 56, pp1060-1093, 1996; Q. Du and P. Gray.

Finite element algorithms and computations for the time-dependent high kappa model for superconductivity, SIAM Sci Comp , R. Kharamikova and J. Peterson.

Ginzburg-Landau vortices: dynamics, pinning and hysteresis, SIAM Math. Anal,
1997, Q. Du and F.-H. Lin.

Analysis and computation of a mean field model for superconductivity, Numer Math; Q. Du, M. Gunzburger, and H.K. Lee. 1999.

Convergence of a numerical method for a mean field model for superconducting vortices, SIAM Numer Anal; Q. Du

Discrete Gauge invariant approximations for the time dependent Ginzburg-Landau models, Math Comp; Q. Du, 1998

Analysis and approximation of the Ginzburg-Landau model of superconductivity

Q. Du, M. Gunzburger and J. Peterson

We consider the Ginzburg-Landau model for superconductivity. We first review some well-known features of superconducting materials and then derive various results concerning the model, the resultant differential equations, and their solution on bounded domains. Then, we consider finite element approximations of the solutions of the Ginzburg-Landau equations and derive error estimates of optimal order.

Solving the Ginzburg-Landau equations by finite element methods

Q. Du, M. Gunzburger and J. Peterson

We consider finite element methods for the approximation of solutions of the Ginzburg-Landau equations of superconductivity. The methods are based on a discretization of the Euler-Lagrange equations resulting from the minimization of the free energy functional. The discretization is effected by requiring the approximate solution to be a piecewise polynomial with respect to a grid. The magnetization versus magnetic field curves obtained through the finite element methods agree well with analogous calculations obtained by other schemes. We demonstrate, both by analyzing the algorithms and through computational experiments, that finite element methods can be very effective and efficient means for the computational simulation of superconducting phenomena and therefore could be applied to determine macroscopic properties of inhomogeneous, anisotropic superconductors.

Finite element approximation of a periodic Ginzburg-Landau model for type-II superconductors]

Q. Du, M. Gunzburger and J. Peterson

We consider efficient finite element algorithms for the computational simulation of type-II superconductors. The algorithms are based on discretizations of a periodic Ginzburg-Landau model. Periodicity is defined with respect to a non-orthogonal lattice that is not necessarily aligned with the coordinate axes; also, the primary dependent variables employed in the model satisfy non-standard quasi''-periodic boundary conditions. After introducing the model, we define finite element schemes, derive error estimates of optimal order, and present the results of some numerical calculations. For a similar quality of simulation, the resulting algorithms seem to be significantly less costly than are previously used numerical approximation methods.

Modeling and analysis of a periodic Ginzburg-Landau model for type-II superconductors

Q. Du, M. Gunzburger and J. Peterson

We consider a periodic Ginzburg-Landau model for superconductivity. The model has two novel features compared to periodic problems arising in other settings. First, periodicity is defined with respect to a non-orthogonal lattice that is not necessarily aligned with the coordinate axes. Second, the periodicity of the physical variables implies non-standard, in the context of periodic problems, relations for the primary dependent variables employed in the model. We introduce the physical assumptions that form the basis for the model and then derive the mathematical model from these assumptions. The model we discuss includes, as special cases, periodic Ginzburg-Landau models appearing in the literature. We then analyze the model equations and its solutions, addressing, among others, questions of existence and regularity. We close with some remarks relevant to the use of the model in conjunction with analytic or numerical approximation methods.

A model for superconducting thin films having variable thickness

Q. Du and M. Gunzburger

A two-dimensional macroscopic model for superconductivity in thin films having variable thickness is derived through an averaging process across the film thickness. The resulting model is similar to the well-known Ginzburg-Landau equations for homogeneous, isotropic materials, except that a function that describes the variations in the thickness of the film now appears in the coefficients of the differential equations. Some results about solutions of the variable thickness model are then given, including existence of solutions and boundedness of the order parameter. It is also shown that the model is consistent in the sense that solutions obtained from the new model are an appropriate limit of a sequence of averages of solutions of the three-dimensional Ginzburg-Landau model as the thickness of the film tends to zero. An application of the variable thickness thin film model to flux pinning is then provided. In particular, the results of numerical calculations are given that show that the vortex-like structures that are present in certain superconductors are attracted to relatively thin regions in a material sample. Finally, extensions of the model to other settings are discussed.

On the Lawrence-Doniach and anisotropic Ginzburg-Landau models for layered superconductors

S. Chapman, Q. Du and M. Gunzburger

We consider two models, the Lawrence-Doniach and the anisotropic Ginzburg-Landau models, for layered superconductors such as the recently discovered high-temperature superconductors. We give a mathematical description of both models and derive existence results for their solution. We then relate the two models in the sense that we show that as the layer spacing tends to zero, the Lawrence-Doniach model reduces to the anisotropic Ginzburg-Landau model. Finally, we derive simplified versions of the models that can be used to accurately simulate high-temperature superconductors.

Computational simulation of type-II superconductivity including pinning phenomena

Q. Du, M. Gunzburger and J. Peterson

A flexible tool, based on the finite element method, for the computational simulation of vortex phenomena in type-II superconductors has been developed. These simulations use refined or newly developed phenomenological models including a time dependent Ginzburg-Landau model, a variable thickness thin film model, simplified models valid for high values of the Ginzburg-Landau parameter, models that account for normal inclusions and Josephson effects, and the Lawrence-Doniach model for layered superconductors. Here, sample results are provided for the case of constant applied magnetic fields. Included in the results are cases of flux pinning by impurities and by thin regions in films.

A Ginzburg-Landau type model of superconducting/normal junctions including Josephson junctions

S. Chapman, Q. Du and M. Gunzburger

A model for superconductors co-existing with normal materials is presented. The model, which applies to such situations as superconductors containing normal impurities and superconductor/normal material junctions, is based on a generalization of the Ginzburg-Landau model for superconductivity. After presenting the model, it is shown that it reduces to well-known models due to de Gennes for certain superconducting/normal interfaces, and in particular, for Josephson junctions. A provident feature of the modified model is that it can, by itself, account for all of these as well as other physical situations. The results of some preliminary computational experiments using the model are then provided; these include flux pinning by normal impurities and a superconductor/normal/superconductor junction. A side benefit of the modified model is that, through its use, these computational simulations are more easily obtained.

Simplified Ginzburg-Landau type models of superconductivity in the high kappa, high field limit

S. Chapman, Q. Du, M. Gunzburger and J. Peterson

A formal asymptotic expansion is used to simplify the Ginzburg-Landau model of superconductivity in the limit of large values of the Ginzburg-Landau parameter and high applied magnetic field strengths. The convergence of solutions of the full Ginzburg-Landau equations to solutions of the leading order equations in the hierarchy is demonstrated for both boundary value and periodic problems. The results of computational experiments using the full Ginzburg-Landau model and the leading order model are given. These indicate that the simplified model yields solutions that are accurate approximations to solutions of the full model equations even for moderate values of the Ginzburg-Landau parameter.

A model for variable thickness superconducting thin films

S. Chapman, Q. Du and M. Gunzburger

A model for superconductivity in thin films having variable thickness is derived through an averaging process across the film. When the film is of uniform thickness the model is identical to a model for superconducting cylinders as the Ginzburg-Landau parameter tends to infinity. This means that all superconducting materials, whether type I or type II in bulk, behave as type-II superconductors when made into sufficiently thin films. When the film is of non-uniform thickness the variations in thickness appear as spatially varying coefficients in the thin-film differential equations. After providing a formal derivation of the model, some results about solutions of the variable thickness model are given. In particular, it is shown that solutions obtained from the new model are an appropriate limit of a sequence of averages of solutions of the three-dimensional Ginzburg-Landau model as the thickness of the film tends to zero. An application of the variable thickness thin film model to flux pinning is then provided. In particular, the results of a numerical calculation are given that show that the vortex-like structures present in superconductors are attracted to relatively thin regions.

Analysis and approximation of optimal control problems for a simplified Ginzburg-Landau model of superconductivity

M. Gunzburger, L. Hou and S. Ravindran

This paper is concerned with optimal control problems for a Ginzburg-Landau model of superconductivity that is valid for high values of the Ginzburg-Landau parameter and high external fields. The control is of Neumann type. We first show that optimal solutions exist. We then show that Lagrange multipliers may be used to enforce the constraints and derive an optimality system from which optimal states and controls may be deduced. Then we define finite element approximations of solutions for the optimality system and derive error estimates for the approximations. Finally, we report on some numerical results.

The critical temperature and gap solution of the Bardeen-Cooper-Schrieffer theory of superconductivity

Q. Du and Y. Yang

Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau equations in superconductivity

Q. Du.

We consider the initial-boundary value problems of the time-dependent nonlinear Ginzburg-Landau equations in superconductivity. It is assumed that the material sample occupies a bounded domain in two and three dimensional spaces. We illustrate that the original equations are not well-posed. In order to fix the lack of uniqueness of the solutions, possible choices of the gauge are identified. Global existence and uniqueness of solutions are proved in a proper gauge. A by-product is the convergence of finite-dimensional Galerkin approximations which may be used in the numerical study of superconductivity phenomena.

High-kappa limit of the time dependent Ginzburg-Landau model for superconductivity

Q. Du and P. Gray.

The time-dependent Ginzburg Landau equations are examined in the high-kappa, high magnetic field setting. Formal asymptotic expansions yield a simplified system of leading-order equations which can be used in the study of vortex motion. The formal asymptotic expansion is then justified by showing the solution to the full time-dependent Ginzburg Landau equations converges to the solution of the leading-order equations as $\kappa\to\infty$. Computational results are also given which show that the simplified leading-order model is indeed an accurate approximation to the solution of the full system of equations for moderate values of kappa.

Finite element algorithms and computations for the time-dependent high kappa model for superconductivity

R. Kharamikova and J. Peterson.

Ginzburg-Landau vortices: dynamics, pinning and hysteresis

Q. Du and F.-H. Lin.

In this paper, we consider three problems related to the mathematical study of vortex phenomena in superconductivity based on the Ginzburg-Landau models. First we study the long time behavior of the solutions of the time-dependent Ginzburg-Landau equations. Then we describe results concerning the the pinning effect of thin regions in a variable thickness thin film. Finally, we prove the existence of vortex like solutions to the steady state Ginzburg-Landau equations and study the hysteresis phenomenon near the lower critical field.

Analysis and computation of a mean field model for superconductivity

Q. Du, M. Gunzburger, and H.K. Lee.

A mean-field model for superconductivity is studied from both the analytical and computational points of view. In this model, the individual vortex-like structures occuring in practical superconductors are not resolved. Rather, these structures are homogenized and a vortex density is solved for. The particular model studies includes effects due to the pinning of vortices. The existence and uniqueness of solutions of a regularized version of the model are demonstrated and the behavior of these solutions as the regularization parameter tends to zero is examined. Then, semi-discrete and fully discrete finite element based discretizations are formulated and analyzed and the results of some computational experiments are pressented.

Convergence of a numerical method for a mean field model for superconducting vortices

Q. Du

In this paper, we analyse the convergence of a finite element/finite volume upwinding scheme for a two-dimensional mean field model of superconducting vortices.

Discrete Gauge invariant approximations for the time dependent Ginzburg-Landau models

Q. Du

We present the mathematical theory for the discrete gauge invariant difference approximations of the time dependent Ginzburg-Landau models.

### Pictures1

Time dependent Ginzburg-Landau model

Steady-state configuration of vortices in a constant applied field in a two-dimensional sample (Contour lines of the magnitude of the order parameter and vector plot of the supercurrent distribution.) Away from the boundary, the vortices arrange themselves in a regular hexagonal lattice.

Ginzburg-Landau parameter: 5
sample size: 20 by 20 coherence lengths
applied magnetic field: 0.5\kappa

### Movie1

Time dependent Ginzburg-Landau model

Nucleation of vortices and development of the Abrikosov vortex lattice in a two-dimensional lattice in a constant applied field. Away from the boundary, the vortices arrange themselves in a regular hexagonal lattice.

Ginzburg-Landau parameter: 5
sample size: 30 by 30 coherence lengths
applied field perpendicular to sample: 0.5\kappa

Movie1 [2 MB]

### Movie2

Time dependent Ginzburg-Landau model

Vortex motion in the presence of an applied current and magnetic field. Vortices move in direction perpendicular to the applied current.

Ginzburg-Landau parameter: 5
sample size: 20 by 20 coherence lengths
applied magnetic field (perpendicular to sample): 0.5\kappa
applied current is in vertical direction

Movie2 [1.2 MB]

### Movie3

Time dependent Ginzburg-Landau model

Anhialation of vortices in the presence of a linearly vaying magnetic field. Vortices of opposite sign move in opposite directions and annihilate each other.

Ginzburg-Landau parameter: 5
sample size: 15 by 15 coherence lengths
applied magnetic field varies linearly from left to right

Movie3 [2.9 MB]

### Pictures2

Variable thickness thin film model.

Ginzburg-Landau parameter: 5
sample size: 20 by 20 coherence lengths
applied magnetic field: 0.5\kappa

Vortices in a constant thickness thin film form a regular lattice (away form boundaries).

Distribution of thin regions of thickness 50% of rest of sample.

Vortices pinned by thin regions in the film and the lattice is distorted.

### Pictures3

Models for high values of the Ginzburg-Landau parameter

Vortex configurations for different values of the Ginzburg-Landau parameter. For \kappa > 5, the vortex configuration is identical to that of the high-\kappa model.

\kappa: 3, 5, 10, 20 (using full Ginzburg-Landau model)

\infty (using high-kappa model)

Effect of sample size on vortex configuration (computed using the high-kappa model). As the sample size is increased, more vortices appear in the sample.

applied magnetic field: 0.5\kappa
sample sizes: 10 by 10, 20 by 20, and 30 by 30 coherence lengths
 Picture7 Picture6 Picture8

Effect of applied magnetic field on vortex configuration (computed using the high-kappa model). As the magnetic field is increased, more vortices appear in the sample.
sample size: 20 by 20 coherence lengths
applied magnetic fields: 0.3\kappa, 0.5\kappa, and 0.7\kappa

### Movies4

Laurence-Doniach model

Vortices in five layers (plotted side by side) with a skewed magnetic field. VOrtices in the different layers align themselves with the field. (Note: the layers should be stacked above each other.)

Ginzburg-Landau parameter: 5
sample size: 13 by 13 coherence lengths
applied magnetic field: (.35, .35, .35)\kappa

Movie4 [1.1 MB]

### Movie5

Laurence-Doniach model

Vortex motion in five layers (plotted side by side) in a skewed magnetic field and an applied current. Vortices move in a direction perpendicular to the applied current. (Note: the layers should be stacked above each other.)

Ginzburg-Landau parameter: 5
sample size: 10 by 20 coherence lengths
applied magnetic field: (.4, .2, .4)\kappa

Movie5 [1.1 MB]

### Pictures4

Anisotropic Ginzburg-Landau model

sample size: 20 by 20 coherence lengths
applied magnetic field (perpendicular to sample): 0.5\kappa Position of grain boundary.
Vortex distribution in presence of grain boundary; vortices are pinned by the grain boundary.

### Pictures5

Models accounting for normal inclusions

Ginzburg-Landau parameter: 5
sample size: 20 by 20 coherence lengths
applied magnetic field: 0.5\kappa

Vortices in homogeneous sample.

Distribution of normal inclusions.

Vortices pinned by normal inclusions.

A 1-coherehnce length thick SNS (superconductor-normal-superconductor) junction.

Vortex distribution in a 1-coherehnce length thick SNS junction.

Supercurrent distribution in a 1-coherehnce length thick SNS junction.

A 2-coherence length thick SNS (superconductor-normal-superconductor) junction.

Vortex distribution in a 2-coherence length thick SNS junction.

Supercurrent distribution in a 2-coherence length thick SNS junction.

A 4-coherence length thick SNS (superconductor-normal-superconductor) junction.

Vortex distribution in a 4-coherence length thick SNS junction.

Supercurrent distribution in a 4-coherence length thick SNS junction.

An 8-coherence length thick SNS (superconductor-normal-superconductor) junction.

Vortex distribution in an 8-coherence length thick SNS junction.

Supercurrent distribution in an 8-coherence length thick SNS junction.

### Movies6

Models accounting for normal inclusions

Ginzburg-Landau parameter: 5
sample size: 20 by 20 coherence lengths
applied magnetic field: 0.5\kappa

Nucleation of vortices and pinning of vortices by normal inclusions in a superconducting sample.
Movie6 [3.9 MB]

### Movies7

Models accounting for normal inclusions

sample size: 20 by 20 coherence lengths
applied magnetic field: 0.5\kappa

Nucleation of vortices in a 1-coherence length thick SNS junction.
Movie7 [1.1 MB]

### Movies8

Models accounting for normal inclusions

sample size: 20 by 20 coherence lengths
applied magnetic field: 0.5\kappa

Nucleation of vortices in a 2-coherence length thick SNS junction.
Movie8 [644K]

### Movies9

Models accounting for normal inclusions

sample size: 20 by 20 coherence lengths
applied magnetic field: 0.5\kappa

Nucleation of vortices in a 4-coherence length thick SNS junction.
Movie9 [677K]

### Movies10

Models accounting for normal inclusions

Nucleation of vortices in an 8-coherence length thick SNS junction.
Movie10 [461K]

### Movies11

Models accounting for thermal fluctuations

Nucleation of vortices and lattice formation in the presence of thermal fluctuations.
Movie11 [4.1 MB]

 Mathematics Department Homepage Faculty Department Research Last updated: 1/20/97