Surface Nucleation of Superconductivity in 3-Dimensions

Lu, Kening; Pan, Xing‐Bin

doi:10.1006/jdeq.2000.3892

Cited by 73 publications

(93 citation statements)

References 18 publications

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“…More recently, a mathematical analysis of these problems has been carried out by many mathematicians. Among them we would like to mention the work of Chapman [C] and Bernoff-Sternberg [BS] based on the formal analysis, Bauman-Phillips-Tang [BPT] for the rigorous analysis on disks, GiorgiPhillips [GP], , [LP2], [LP3], [LP4], [LP5], [LP6], del Pino-FelmerSternberg [DFS], Pan [P], Helffer-Morame [HMor] and Helffer-Pan [HP] for rigorous analysis on general domains.…”

Section: §1 Introductionmentioning

confidence: 99%

“…To study the nucleation of superconductivity for a sample with large value of the Ginzburg-Landau parameter κ, and subject to a strong magnetic field H appl ≡ H = σH 0 with large σ, Lu and Pan [LP4], [LP5] introduced a number σ * (κ), which depends on H 0 , such that the sample is in the normal state if σ > σ * (κ) and is in the superconducting state if 0 ≤ σ < σ * (κ). In the special case when the applied field is homogeneous, σ * (κ) is equal to the upper-critical field H C3 .…”

Section: §1 Introductionmentioning

confidence: 99%

“…In the special case where the applied field is homogeneous, the above equality yields an estimate for H C3 . Although in [LP1], [LP2], [LP3], [LP4], [LP5] the authors mainly discussed the case where the applied field does not vanish, it was observed in [LP3] (Proposition 6.3) that the value of σ * (κ) is greatly raised if H 0 has zeros: σ * (κ) ≥ Cκ 2 for all large κ. In this paper we shall study this phenomenon more precisely.…”

Section: §1 Introductionmentioning

confidence: 99%

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Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains

Pan

Kwek

2002

Trans. Amer. Math. Soc.

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Abstract. We establish an asymptotic estimate of the lowest eigenvalue µ(bF) of the Schrödinger operator −∇ 2 bF with a magnetic field in a bounded 2-dimensional domain, where curl F vanishes non-degenerately, and b is a large parameter. Our study is based on an analysis on an eigenvalue variation problem for the Sturm-Liouville problem. Using the estimate, we determine the value of the upper critical field for superconductors subject to non-homogeneous applied magnetic fields, and localize the nucleation of superconductivity. §1. IntroductionIn this paper we study the lowest eigenvalue µ(bF) for the Schrödinger operator −∇ 2 bF with a magnetic field F, where curl F vanishes non-degenerately in a bounded 2-dimensional domain, and derive asymptotic estimates for the large parameter b. This problem arises in the mathematical theory of describing the nucleation phenomenon for superconductors subject to non-homogeneous applied magnetic fields. Using the estimates of µ(bF), we determine the value of the upper critical field and the location of nucleation of superconductivity. This problem is interesting to us also because of its connection to the problem of estimating the lowest eigenvalue and describing the (bounded) eigenfunctions 1 of the Schrödinger operator with a non-degenerately vanishing magnetic field in the entire plane R 2 and in the half-plane R 2 + . A type 2 superconductor subject to an applied magnetic field will exhibit many interesting phenomena. It is well-known that, if the applied magnetic field is homogeneous and decreases from the upper critical value H C3 , superconductivity nucleates at the surface of the sample. The estimate of the upper critical field and the localization of the nucleation of superconductivity have been studied by many physicists; see Saint-James and De Gennes Key words and phrases. Schrödinger operator with a magnetic field, eigenvalue, GinzburgLandau system, superconductivity, nucleation, upper critical field, Sturm-Liouville operator, Riccati type equation.1 In this paper, for convenience, when considering eigenvalue problems in the entire plane or in the half-plane, we call a non-trivial bounded solution an eigenfunction. Therefore, the eigenfunctions associated with the lowest eigenvalue need not lie in L 2 .

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Section: §1 Introductionmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

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Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains

Pan

Kwek

2002

Trans. Amer. Math. Soc.

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“…In this and the following subsections we will use the non-existence results from Subsection 4.1 to obtain improved versions of the estimates in Theorem 3.1 in a reduced parameter range. The application of this idea ('blow-up') to the GinzburgLandau system appeared to our knowledge first in [LuPa1,LuPa2] and has since been used extensively since (see for instance [LuPa4,Pan4,HePa]). …”

Section: Asymptotic Estimatesmentioning

confidence: 99%

“…In the literature such estimates are found in varying generality scattered over different publications (c.f. [LuPa1,LuPa2,LuPa3,LuPa4], [HePa],...).…”

Section: Introductionmentioning

confidence: 99%

Optimal uniform elliptic estimates for the Ginzburg-Landau system

Fournais¹,

Helffer²

2007

Adventures in Mathematical Physics

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On the Ginzburg‐Landau critical field in three dimensions

Fournais

Helffer

2008

Comm Pure Appl Math

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Abstract. We study the three dimensional Ginzburg-Landau model of superconductivity. Several 'natural' definitions of the (third) critical field, H C 3 , governing the transition from the superconducting state to the normal state, are considered. We analyze the relation between these fields and give conditions as to when they coincide. An interesting part of the analysis is the study of the monotonicity of the ground state energy of the Laplacian, with constant magnetic field and with Neumann (magnetic) boundary condition, in a domain Ω. It is proved that the ground state energy is a strictly increasing function of the field strength for sufficiently large fields. As a consequence of our analysis we give an affirmative answer to a conjecture by Pan.

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Surface Nucleation of Superconductivity in 3-Dimensions

Cited by 73 publications

References 18 publications

Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains

Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains

Optimal uniform elliptic estimates for the Ginzburg-Landau system

On the Ginzburg‐Landau critical field in three dimensions

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