Abstract
We study a 3D Ginzburg-Landau model in a half-space which is expected to capture the key features of surface superconductivity for applied magnetic fields between the second critical field \(H_{C_{2}}\) and the third critical field \(H_{C_{3}}\). For the magnetic field in this regime, it is known from physics that superconductivity should be essentially restricted to a thin layer along the boundary of the sample. This leads to the introduction of a Ginzburg-Landau model on a half-space. We prove that the non-linear Ginzburg-Landau energy on the half-space with constant magnetic field is a decreasing function of the angle ν that the magnetic field makes with the boundary. In the case when the magnetic field is tangent to the boundary (ν = 0), we show that the energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. For non-parallel applied fields, we also construct a periodic problem with vortex lattice minimizers reproducing the effective energy, which suggests that the order parameter of the full Ginzburg-Landau model will exhibit 3 dimensional vortex structure near the surface of the sample.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abrikosov, A.: On the magnetic properties of superconductors of the second type. Sov. Phys. JETP 5, 1174–1182 (1957)
Aftalion, A., Serfaty, S.: Lowest Landau level approach in superconductivity for the Abrikosov lattice close to \(H_{C_{2}}\). Sel. Math. (N.S.) 13, 183–202 (2007)
Baldo, S., Jerrard, R.L., Orlandi, G., Soner, H.M.: Convergence of Ginzburg-Landau functionals in three-dimensional superconductivity. Arch. Ration. Mech. Anal. 205(3), 699–752 (2012)
Bonnaillie-Noël, V., Dauge, M., Popoff, N., Raymond, N.: Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions. Z. Angew. Math. Phys. 63, 203–231 (2012)
Correggi, M., Rougerie, N.: On the Ginzburg-Landau functional in the surface superconductivity regime. Comm. Math. Phys. 332, 1297–1343 (2014)
Correggi, M., Rougerie, N.: Effects of boundary curvature on surface superconductivity. Lett. Math. Phys. 106(4), 445–467 (2016)
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)
de Gennes, P.G.: Superconductivity of metals and alloys. Benjamin, New York (1966)
Fournais, S., Helffer, B.: Spectral Methods in Surface Superconductivity Progress in Nonlinear Differential Equations and Their Applications, vol. 77. Birkhäuser Boston Inc, Boston (2010)
Fournais, S., Helffer, B., Persson, M.: Superconductivity between \(H_{C_{2}}\) and \(H_{C_{3}}\). J. Spectr. Theory 1, 273–298 (2011)
Fournais, S., Kachmar, A.: The ground state energy of the three dimensional Ginzburg-Landau functional Part I: Bulk regime. Comm. Partial Differ. Equ. 38, 339–383 (2013)
Fournais, S., Kachmar, A., Persson, M.: The ground state energy of the three dimensional Ginzburg-Landau functional. Part II: Surface regime. J. Math. Pures Appl. (9) 99, 343–374 (2013)
Frank, R.L., Hainzl, C., Seiringer, R., Solovej, J.P.: Microscopic derivation of Ginzburg-Landau theory. J. Amer. Math. Soc. 25, 667–713 (2012)
Ginzburg, V., Landau, L.: On the theory of superconductivity. Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950)
Gor’kov, L.: Microscopic derivation of the ginzburg-landau equations in the theory of superconductivity. Zh. Eksp. Teor. Fiz. 36, 1918–1923 (1959)
Helffer, B., Morame, A.: Magnetic bottles in connection with superconductivity. J. Funct. Anal. 185, 604–680 (2001)
Helffer, B., Morame, A.: Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case). Ann. Sci. Éc. Norm. Supér. 37(4), 105–170 (2004)
Lu, K., Pan, X.-B.: Gauge invariant eigenvalue problems in r 2 and in \(r^{2}_{+}\). Trans. Amer. Math. Soc. 352, 1247–1276 (2000)
Lu, K.: Surface nucleation of superconductivity in 3-dimensions. J. Differ. Equ. 168, 386–452 (2000). Special issue in celebration of Jack K. Hale’s 70th birthday, Part 2 (Atlanta, GA/Lisbon, 1998)
Pan, X.-B.: Surface superconductivity in applied magnetic fields above \(H_{C_{2}}\). Comm. Math. Phys. 228, 327–370 (2002)
Pan, X.-B.: Surface superconductivity in 3 dimensions. Trans. Amer. Math. Soc. 356, 3899–3937 (2004)
Persson, A.: Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator. Math. Scand. 8, 143–153 (1960)
Raymond, N.: On the semiclassical 3D Neumann Laplacian with variable magnetic field. Asymptot. Anal. 68, 1–40 (2010)
Sandier, E., Serfaty, S.: Vortices in the Magnetic Ginzburg-Landau Model Progress in Nonlinear Differential Equations and Their Applications, vol. 70. Birkhäuser Boston, Inc, Boston (2007)
Acknowledgments
Fournais and Miqueu were partially supported by a Sapere Aude grant from the Independent Research Fund Denmark, Grant number DFF–4181-00221. Pan was partially supported by the National Natural Science Foundation of China grants no. 11671143 and no. 11431005.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fournais, S., Miqueu, JP. & Pan, XB. Concentration Behavior and Lattice Structure of 3D Surface Superconductivity in the Half Space. Math Phys Anal Geom 22, 12 (2019). https://doi.org/10.1007/s11040-019-9307-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-019-9307-7
Keywords
- Ginzburg-Landau equations
- Superconductivity
- Surface superconductivity
- Magnetic Schrodinger operator
- Surface concentration
- 3D vortices
- Partial differential equations
- Calculus of variation
- Estimate of eigenvalue