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The Parr formula for the superheating field in a semi-infinite film
Journal of Mathematical Physics
Abstract
Di Bartolo, Dolgert, and Dorsey [
Phys. Rev. B 53, 5650–5660 (1996)
] have constructed asymptotic matched solutions at order 2 for the half-space Ginzburg–Landau model in the weak-κ limit. These authors deduced a formal expansion for the superheating field hsh(κ) up to order 4, extending the de Gennes formula [
Proceedings of the Eighth Latin American School of Physics, Caracas, 1966
] and the two terms in Parr’s formula [
Z. Phys. B 25, 359–361 (1976)
]. On the other hand, the present author [
Eur. J. Appl. Math 13, 519–547 (2002)
] obtained two terms in the lower bound for hsh(κ). In this paper, we prove rigorously that the second term of the expansion of hsh(κ) is of the order of O(κ1/2) and we get the Parr formula. We improve the upper bound obtained by Bolley and Helffer [
Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, 597–613 (1997)
] and we get κ(hsh(κ))2 ⩽ 2−3/2 = κ+O(κ1+ρ), ρ>0. The proof is based on new estimates for f′, A, and A′. To achieve this, we are guided by the analysis of the properties of the approximate solution constructed previously in [
Del Castillo, Math Modell. Numer. Anal. 36, 971–973 (2002)
;
J. Math. Phys. 44, 2416–2450 (2003)
;
Dolgert et al., Phys. Rev. B 53, 5650–5660 (1996)
].
We have constructed asymptotic matched solutions for the one dimensionnal Ginzburg-Landau system when kappa is small [Math. Model. Num. Anal. 36, 971-993 (2002)]. We have deduced an expansion in powers of kappa1/2 at all orders for the superheating field. In this paper, using these constructions, we propose to show that the superheating field admits for lower bound the expansion of the formal superheating field truncated at order n, for all n∈N. We generalize the proof given in Eur. J. Appl. Math. 13, 519-547 (2002), where this result is obtained for n=1. Then, we construct solutions of the Ginzburg-Landau system when the exterior magnetic field is near to the superheating field, and we give a localization of these solutions.
Based on a shooting alternative that allows one to numerically solve the one-dimensional system of Ginzburg–Landau in an unbounded
domain, a numerical study of the stability of solutions of this system is performed here. This stability notion, from a physical
point of view, means that each solution of the system is identified as stable when it minimizes the corresponding Ginzburg–Landau
functional. As opposed to a previous paper, the present one is concerned with a more general study since the weak and large
regimes of the Ginzburg–Landau parameter are considered and the initial data are no longer subject to the de Gennes condition.
Certain conjectures regarding the superheating field are also investigated numerically.
KeywordsGeneralized eigenvalue problems–Ginzburg–Landau systems–Hermite finite elements–Initial-value problems–Shooting methods–Superconductivity
The superheated Meissner state in type-I superconductors is studied both analytically and numerically within the framework of Ginzburg-Landau theory. Using the method of matched asymptotic expansions we have developed a systematic expansion for the solutions of the Ginzburg-Landau equations in the limit of small , and have determined the maximum superheating field for the existence of the metastable, superheated Meissner state as an expansion in powers of . Our numerical solutions of these equations agree quite well with the asymptotic solutions for . The same asymptotic methods are also used to study the stability of the solutions, as well as a modified version of the Ginzburg-Landau equations which incorporates nonlocal electrodynamics. Finally, we compare our numerical results for the superheating field for large- against recent asymptotic results for large-, and again find a close agreement. Our results demonstrate the efficacy of the method of matched asymptotic expansions for dealing with problems in inhomogeneous superconductivity involving boundary layers. Comment: 14 pages, 8 uuencoded figures, Revtex 3.0
A moderately small superconductor is defined as one that has at least one dimension perpendicular to the applied magnetic field, which is comparable to λ(T) at a temperature T=TL, which is defined as the Landau Critical Point (LCP) temperature. This definition excludes a thin film in an applied magnetic field perpendicular to its surface. The chapter discusses the Ginzburg–Landau (GL) theory that is relatively simple, conceptually, yet rich in its phenomenological insight and powerful in obtaining critical magnetic fields, although not always in a simple manner. This theory is based upon the electrodynamics of superconductors by London and London that, in turn, is based on the two-fluid model by Gorter and Casimir. Gor’kov showed that the GL theory could be derived in the appropriate limits from the microscopic theory, thus providing the basic justification for the GL theory. The GL theory provides a convenient foundation for understanding the basic phenomena of size effects and magnetic phase transitions of superconductors.
The aim of this paper is to analyze numerically the different results concerning the superheating field for the Ginzburg-Landau equations published by the physicists. In the case when the size of the film is large in comparison with the inverse of the characteristic constant κ of the material, we present an approximate model and analyze how it fits with previous numerical results and with our new computations. A rigorous but partial study in the weak κ limit is presented in our other paper [8].
This paper is the first part of a mathematical study of the Ginzburg-Landau equations for a film. We first give here mathematical definitions for the different critical fields. We improve results on the existence and uniqueness of bifurcating superconducting solutions. Then, we analyze different limit problems first introduced by physicists: in particular a limit constant model which will be seen in Part II as the starting point of the study of the asymptotic problem when κ tends to 0 or when the width of the film tends to 0.
We have constructed asymptotic matched solutions for the one dimensionnal Ginzburg-Landau system when kappa is small [Math. Model. Num. Anal. 36, 971-993 (2002)]. We have deduced an expansion in powers of kappa1/2 at all orders for the superheating field. In this paper, using these constructions, we propose to show that the superheating field admits for lower bound the expansion of the formal superheating field truncated at order n, for all n∈N. We generalize the proof given in Eur. J. Appl. Math. 13, 519-547 (2002), where this result is obtained for n=1. Then, we construct solutions of the Ginzburg-Landau system when the exterior magnetic field is near to the superheating field, and we give a localization of these solutions.
In continuation with our preceding paper [10] concerning the superconducting film, we present in this article rigorous results concerning the superheating in the weak kappa limit. The principal result is an important step toward the rigorous proof of a formula due to P. De Gennes [26] . This paper is complementary to our paper [11] where numerical results are presented and approximate models are discussed. Most of the results have been announced in [12] and [13].
We have calculated analytically the superheating fieldH
sh
for bulk superconductors, correct to second order inκ. We find{{H_{sh} } \mathord{\left/ {\vphantom {{H_{sh} } {H_c }}} \right. \kern-\nulldelimiterspace} {H_c }} = (\kappa ^{ - \tfrac{1}{2}} 2^{ - \tfrac{1}{4}} ) \cdot ({{1 + 15\sqrt 2 \kappa } \mathord{\left/ {\vphantom {{1 + 15\sqrt 2 \kappa } {32}}} \right. \kern-\nulldelimiterspace} {32}}), which agrees well with numerical computations forκ<0.5. The surface order parameter isf_0 (H_{sh} ) = ({1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right. \kern-\nulldelimiterspace} {\sqrt 2 }})({{1 - 7\sqrt {2\kappa } } \mathord{\left/ {\vphantom {{1 - 7\sqrt {2\kappa } } {32}}} \right. \kern-\nulldelimiterspace} {32}}), and the penetration depth is{{\lambda (H_{sh} )} \mathord{\left/ {\vphantom {{\lambda (H_{sh} )} {\lambda (0)}}} \right. \kern-\nulldelimiterspace} {\lambda (0)}} = \sqrt 2 ({{1 + 3\sqrt 2 \kappa } \mathord{\left/ {\vphantom {{1 + 3\sqrt 2 \kappa } {32}}} \right. \kern-\nulldelimiterspace} {32}}).
We prove, in this article, the existence and uniqueness of a solution of the Ginzburg-Landau system in the unbounded interval]0, +infinity[ when the value of the order parameter is given at 0. Those results give a proof, for any value of the Ginzburg-Landau parameter kappa, of a conjecture in [7] which expresses a numerical evidence admitted by the physical literature. The uniqueness proof uses the Sturm Comparison Theorem which gives some monotonicity result of the solutions with respect to the exterior magnetic field. The existence comes from the existence of positive solutions for an analogous problem posed in a bounded interval and from the extraction of a converging diagonal sub-sequence.
We study the asymptotic behavior of the local superheating field for a film of width 2d in the regime κ small, κd large, where κ is the Ginzburg–Landau parameter. This gives a mathematical justification for the introduction of the semi-infinite model as a good approximation for this regime. © 2000 American Institute of Physics.
Following our preceding papers [1, 2]
concerning semi-infinite superconducting films, we
consider new a
priori estimates on the exterior magnetic
field h=A′(0), when (f, A)
is a solution of the corresponding Ginzburg–Landau system. The main
new
results concern the limit as
κ[rightward arrow][infty infinity], but we prove also the existence of a finite
superheating field. We also discuss recent
results [3] concerning the superheating field in the large
κ limit, and show how to relate these
formal solutions to suitable subsolutions and supersolutions giving
the existence of a solution for
h<1/[surd radical]2 and κ large enough. We also analyse
the same problem by variational techniques
and get the existence of a locally stable solution for h<1/[surd radical]2
and any κ>0.
Dorsey et
al. [8] have constructed formal solutions for the half-space Ginzburg–Landau
model, when κ is small. Dorsey et
al. deduce a formal expansion for the superheating field
in powers of κ[fraction one-half] up to order 4. In this paper, we show how the formal construction gives
a natural way for constructing a subsolution for the Ginzburg–Landau system. We improve
the result obtained by Bolley & Helffer [2], and take a step in the proof of the Parr Formula
[13], getting two terms in the lower bound for the superheating field as κ [rightward arrow] 0.
In a recent paper devoted to the study of the superheating field attached to a semi-infinite superconductor, Chapman [1] constructs a family of approximate solutions of the Ginzburg–Landau system. This construction, based on a matching procedure, implicitly uses the existence of a family of solutions depending on a parameter c ∈ IR of the Painlevé equation in a semi-infinite interval (0, +∞)
with a Neumann condition at 0
and having a prescribed behaviour at +∞
In this paper we prove the existence of such a family of solutions and investigate its properties. Moreover, we prove that the second coefficient in Chapman's expansion of the superheating field is finite.
This paper presents an algebraic multigrid method for the efficient solution of the linear system arising from a finite element discretization of variational problems in H0(curl,Ω). The finite element spaces are generated by Nédélec's edge elements.
A coarsening technique is presented, which allows the construction of suitable coarse finite element spaces, corresponding transfer operators and appropriate smoothers. The prolongation operator is designed such that coarse grid kernel functions of the curl-operator are mapped to fine grid kernel functions. Furthermore, coarse grid kernel functions are ‘discrete’ gradients. The smoothers proposed by Hiptmair and Arnold, Falk and Winther are directly used in the algebraic framework.
Numerical studies are presented for 3D problems to show the high efficiency of the proposed technique. Copyright
Superconductivity was discovered by Kammerlingh-Onnes in 1911, when John Bardeen was only three years old. Who would have thought that over half a century would be required to develop an understanding of the mechanism of superconductivity and that Bardeen in particular would play the key role in this development? The history of superconductivity is very interesting and even dramatic, and I am surprised how little this subject is elucidated in the literature. I must admit that I myself found time to look through the first papers in this field only in 1979, even though ! started to study the problem of Superconductivity in 1943! At the same time, most of these early articles appeared in the journal Communications from the
Macroscopic theory of superconductivity valid for magnetic fields of arbitrary magnitude and the behaviour of superconductors
in weak high frequency fields are discussed. The problem of formulating a microscopic theory of superconductivity is also
considered.
Si discute una teoria macroscopica della superconduttività valida per campi magnetici di intensità arbitraria e il comportamento
dei superconduttori in campi deboli di alta frequenza. Si considera altresì il problema della formulazione di una teoria microscopica
della superconduttività.
The superheating magnetic field of a type II superconductor is examined, using the time-dependent Ginzburg-Landau equations and the methods of formal asymptotics. The superconducting solution in a halfspace is found to exist only for magnetic fields lower than some critical value where there is a folding over of the solution branch. A linear stability analysis is performed both in one and two dimensions, giving differing criteria for stability. Finally, superheating fields for more general geometries are considered, and in particular the case of a sine-wave perturbation of a halfspace is examined.
Dorsey, Di Bartolo and Dolgert (Di Bartolo et al., 1996; 1997) have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model, in the weak- limit. These authors deduced a formal expansion for the superheating field in powers of up to order four, extending the formula by De Gennes (De Gennes, 1966) and the two terms in Parr's formula (Parr, 1976). In this paper, we construct asymptotic matched solutions at all orders leading to a complete expansion in powers of for the superheating field.
In continuation with our preceding paper [3] concerning the superconducting film, we present in this article new estimates for the superheating field in the weak κ limit. The principal result is the proof of the existence of a finite superheating field h^{sh,+}(κ) (obtained by restricting the usual definition of the superheating field to solutions of the Ginzburg-Landau system (f, A) with f positive) in the case of a semi-infinite interval. The bound is optimal in the limit κ → 0 and permits to prove (combining with our previous results) the De Gennes formula
2^{ - \frac{3}{4}} = \lim_{\kappa \rightarrow 0} \kappa ^{\frac{1}{2}}h^{sh, + }\left(\kappa \right)
The proof is obtained by improving slightly the estimates given in [3] where an upper bound was found but under the additional condition that the function f was bounded from below by some fixed constant ρ > 0 .
Résumé
Poursuivant notre étude précédente [3] sur les films supraconducteurs, nous présentons dans cet article de nouvelles estimations sur le champ de surchauffe lorsque κ tend vers 0 . Le résultat principal est la démonstration, dans le cas d’un intervalle semi-infini, de l’existence d’un champ de surchauffe fini h^{sh,+}(κ) (obtenue en restreignant la définition usuelle du champ de surchauffe aux solutions (f, A) du système de Ginzburg–Landau telles que f soit positive). La majoration est optimale à la limite κ → 0 et permet de montrer (en regroupant avec nos résultats précédents) la formule de De Gennes
2^{ - \frac{3}{4}} = \lim_{\kappa \rightarrow 0}\kappa ^{\frac{1}{2}}h^{sh, + }\left(\kappa \right)
La démonstration est obtenue en améliorant légèrement les estimations de l’article [3] dans lequel nous donnions une majoration, mais en supposant la fonction f minorée par une constante ρ > 0 fixée.
This paper gives a complete description of the solutions of the one dimensional Ginzburg-Landau equations which model superconductivity phenomena in infinite slabs. We investigate this problem over the entire range of physically important parameters: a the size of the slab, the Ginzburg-Landau parameter, and , the exterior magnetic field. We do extensive numerical computations using the software AUTO, and determine the number, symmetry and stability of solutions for all values of the parameters. In particular, our experiments reveal the existence of two key-points in parameter space which play a central role in the formation of the complicated patterns by means of bifurcation phenomena. Our global description also allows us to separate the various physically important regimes, to classify previous results in each regime according to the values of the parameters and to derive new open problems. In addition, our investigation provides new insight into the problem of differentiating between the types of superconductors in terms of the parameters. Comment: 36 pages, 15 figures. to appear in physica D
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