Abstract
We investigate, within 2D Ginzburg-Landau theory, the ground state of a type-II superconducting cylinder in a parallel magnetic field varying between the second and third critical values. In this regime, superconductivity is restricted to a thin shell along the boundary of the sample and is to leading order constant in the direction tangential to the boundary. We exhibit a correction to this effect, showing that the curvature of the sample affects the distribution of superconductivity.
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References
Abrikosov A.: On the magnetic properties of superconductors of the second type. Soviet Phys. JETP. 5, 1174–1182 (1957)
Almog Y.: Nonlinear Surface Superconductivity in the Large \({\kappa}\) Limit. Rev. Math. Phys. 16, 961–976 (2004)
Almog Y.: Abrikosov lattices in finite domains. Commun. Math. Phys. 262, 677–702 (2006)
Almog Y., Helffer B.: The distribution of surface superconductivity along the boundary: on a conjecture of X.B. Pan. SIAM J. Math. Anal. 38, 1715–1732 (2007)
Béthuel, F., Brézis, H., Hélein, F.: Ginzburg-Landau Vortices, Progress in nonlinear differential equations and their applications, vol. 13. Birkhäuser, Basel (1994)
Correggi, M., Rougerie, N.: On the Ginzburg-Landau functional in the surface superconductivity regime. Commun. Math. Phys. 332, 1297–1343 (2014). [Erratum ibid. 338, 1451–1452 (2015)]
Correggi M., Rougerie N.: Boundary behavior of the Ginzburg-Landau order parameter in the surface superconductivity regime. Arch. Ration. Mech. Anal. 219(1), 553–606 (2016)
Fournais S., Helffer B.: Energy asymptotics for type II superconductors. Calc. Var. Partial Differ. Equ. 24, 341–376 (2005)
Fournais S., Helffer B.: On the third critical field in Ginzburg-Landau theory. Commun. Math. Phys. 266, 153–196 (2006)
Fournais, S., Helffer, B.: Spectral Methods in Surface Superconductivity. In: Progress in nonlinear differential equations and their applications, vol. 77. Birkhäuser, Basel (2010)
Fournais S., Helffer B., Persson M.: Superconductivity between \({H_{c2}}\) and \({H_{c3}}\). J. Spectr. Theory. 1, 273–298 (2011)
Fournais S., Kachmar A.: Nucleation of bulk superconductivity close to critical magnetic field. Adv. Math. 226, 1213–1258 (2011)
Fournais S., Kachmar A., Persson M.: The ground state energy of the three dimensional Ginzburg-Landau functional. Part II: surface regime. J. Math. Pures. App. 99, 343–374 (2013)
Kachmar A.: The Ginzburg-Landau order parameter near the second critical field. SIAM J. Math. Anal. 46, 572–587 (2014)
Lu K., Pan X.B.: Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity. Phys. D. 127, 73–104 (1999)
Nirenberg L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa. 13, 115–162 (1959)
Pan X.B.: Surface superconductivity in applied magnetic fields above \({H_{\rm c2}}\). Commun. Math. Phys. 228, 327–370 (2002)
Sandier, E., Serfaty, S.:Vortices in the Magnetic Ginzburg-Landau Model. In: Progress in nonlinear differential equations and their applications, vol. 70. Birkhäuser, Basel (2007)
Sigal, I.M.: Magnetic vortices, abrikosov lattices and automorphic functions. (2013). arXiv:1308.5446
Saint-James D., de Gennes P.G.: Onset of superconductivity in decreasing fields. Phys. Lett. 7, 306–308 (1963)
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Correggi, M., Rougerie, N. Effects of Boundary Curvature on Surface Superconductivity. Lett Math Phys 106, 445–467 (2016). https://doi.org/10.1007/s11005-016-0824-z
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DOI: https://doi.org/10.1007/s11005-016-0824-z