November  2019, 24(11): 5903-5926. doi: 10.3934/dcdsb.2019112

Analysis of minimizers of the Lawrence-Doniach energy for superconductors in applied fields

1. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

2. 

Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA

* Corresponding author: Guanying Peng

Received  September 2018 Published  June 2019

Fund Project: The authors were supported in part by NSF Grant DMS-1109459

We analyze minimizers of the Lawrence-Doniach energy for layered superconductors with Josephson constant $ \lambda $ and Ginzburg-Landau parameter $ 1/\epsilon $ in a bounded generalized cylinder $ D = \Omega\times[0, L] $ in $ \mathbb{R}^3 $, where $ \Omega $ is a bounded simply connected Lipschitz domain in $ \mathbb{R}^2 $. Our main result is that in an applied magnetic field $ \vec{H}_{ex} = h_{ex}\vec{e}_{3} $ which is perpendicular to the layers with $ \left|\ln\epsilon\right|\ll h_{ex}\ll\epsilon^{-2} $, the minimum Lawrence-Doniach energy is given by $ \frac{|D|}{2}h_{ex}\ln\frac{1}{\epsilon\sqrt{h_{ex}}}(1+o_{\epsilon, s}(1)) $ as $ \epsilon $ and the interlayer distance $ s $ tend to zero. We also prove estimates on the behavior of the order parameters, induced magnetic field, and vorticity in this regime. Finally, we observe that as a consequence of our results, the same asymptotic formula holds for the minimum anisotropic three-dimensional Ginzburg-Landau energy in $ D $ with anisotropic parameter $ \lambda $ and $ o_{\epsilon, s}(1) $ replaced by $ o_{\epsilon}(1) $.

Citation: Patricia Bauman, Guanying Peng. Analysis of minimizers of the Lawrence-Doniach energy for superconductors in applied fields. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5903-5926. doi: 10.3934/dcdsb.2019112
References:
[1]

S. AlamaA. J. Berlinsky and L. Bronsard, Minimizers of the Lawrence-Doniach energy in the small-coupling limit: Finite width samples in a parallel field, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 281-312. doi: 10.1016/S0294-1449(01)00081-6. Google Scholar

[2]

S. AlamaL. Bronsard and A. J. Berlinsky, Periodic vortex lattices for the Lawrence-Doniach model of layered superconductors in a parallel field, Commun. Contemp. Math., 3 (2001), 457-494. doi: 10.1142/S0219199701000457. Google Scholar

[3]

S. AlamaL. Bronsard and E. Sandier, On the shape of interlayer vortices in the Lawrence-Doniach model, Trans. Amer. Math. Soc., 360 (2008), 1-34. doi: 10.1090/S0002-9947-07-04188-8. Google Scholar

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S. AlamaL. Bronsard and E. Sandier, On the Lawrence-Doniach model of superconductivity: Magnetic fields parallel to the axes, J. Eur. Math. Soc., 14 (2012), 1825-1857. doi: 10.4171/JEMS/348. Google Scholar

[5]

S. AlamaL. Bronsard and E. Sandier, Minimizers of the Lawrence-Doniach functional with oblique magnetic fields, Comm. Math. Phys., 310 (2012), 237-266. doi: 10.1007/s00220-011-1399-2. Google Scholar

[6]

G. AlbertiS. Baldo and G. Orlandi, Variational convergence for functionals of Ginzburg-Landau type, Indiana Univ. Math. J., 54 (2005), 1411-1472. doi: 10.1512/iumj.2005.54.2601. Google Scholar

[7]

S. BaldoR. L. JerrardG. Orlandi and H. M. Soner, Convergence of Ginzburg-Landau functionals in three-dimensional superconductivity, Arch. Rational Mech. Anal., 205 (2012), 699-752. doi: 10.1007/s00205-012-0527-2. Google Scholar

[8]

S. BaldoR. L. JerrardG. Orlandi and H. M. Soner, Vortex density models for superconductivity and superfluidity, Comm. Math. Phys., 318 (2013), 131-171. doi: 10.1007/s00220-012-1629-2. Google Scholar

[9]

P. Bauman and Y. Ko, Analysis of solutions to the Lawrence-Doniach system for layered superconductors, SIAM J. Math. Anal., 37 (2005), 914-940. doi: 10.1137/S0036141004444597. Google Scholar

[10]

S. J. ChapmanQ. Du and M. D. Gunzburger, On the Lawrence-Doniach and anisotropic Ginzburg-Landau models for layered superconductors, SIAM J. Appl. Math., 55 (1995), 156-174. doi: 10.1137/S0036139993256837. Google Scholar

[11]

E. B. FabesM. Jr Jodeit and N. M. Rivière, Potential techniques for boundary value problems on ${C}^1$-domains, Acta Math., 141 (1978), 165-186. doi: 10.1007/BF02545747. Google Scholar

[12]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, reprint of the 1998 ed., Springer-Verlag, Berlin, 2001. Google Scholar

[13]

T. Giorgi and D. Phillips, The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model, SIAM Review, 44 (2002), 237-256. doi: 10.1137/S003614450139951. Google Scholar

[14]

Y. Iye, How anisotropic are the cuprate high Tc superconductors?, Comments Cond. Mat. Phys., 16 (1992), 89-111. Google Scholar

[15]

R. L. Jerrard and H. M. Soner, Limiting behavior of the Ginzburg-Landau functional, J. Funct. Anal., 192 (2002), 524-561. doi: 10.1006/jfan.2001.3906. Google Scholar

[16]

A. Kachmar, The ground state energy of the three-dimensional Ginzburg-Landau model in the mixed phase, J. Funct. Anal., 261 (2011), 3328-3344. doi: 10.1016/j.jfa.2011.08.002. Google Scholar

[17]

G. Peng, Convergence of the Lawrence-Doniach energy for layered superconductors with magnetic fields near $H_c_1$, SIAM J. Math. Anal., 49 (2017), 1225-1266. doi: 10.1137/16M1064398. Google Scholar

[18]

E. Sandier and S. Serfaty, On the energy of type-II superconductors in the mixed phase, Rev. Math. Phys., 12 (2000), 1219-1257. doi: 10.1142/S0129055X00000411. Google Scholar

[19]

E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity, Ann. Sci. École Norm. Sup., 33 (2000), 561–592. doi: 10.1016/S0012-9593(00)00122-1. Google Scholar

[20]

E. Sandier and S. Serfaty, The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model, SIAM J. Math. Anal., 34 (2003), 939-956. doi: 10.1137/S0036141002406084. Google Scholar

[21]

E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Progress in Nonlinear Differential Equations and Their Applications 70, Birkhäuser, Boston, 2007. Google Scholar

[22]

G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal., 59 (1984), 572-611. doi: 10.1016/0022-1236(84)90066-1. Google Scholar

show all references

References:
[1]

S. AlamaA. J. Berlinsky and L. Bronsard, Minimizers of the Lawrence-Doniach energy in the small-coupling limit: Finite width samples in a parallel field, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 281-312. doi: 10.1016/S0294-1449(01)00081-6. Google Scholar

[2]

S. AlamaL. Bronsard and A. J. Berlinsky, Periodic vortex lattices for the Lawrence-Doniach model of layered superconductors in a parallel field, Commun. Contemp. Math., 3 (2001), 457-494. doi: 10.1142/S0219199701000457. Google Scholar

[3]

S. AlamaL. Bronsard and E. Sandier, On the shape of interlayer vortices in the Lawrence-Doniach model, Trans. Amer. Math. Soc., 360 (2008), 1-34. doi: 10.1090/S0002-9947-07-04188-8. Google Scholar

[4]

S. AlamaL. Bronsard and E. Sandier, On the Lawrence-Doniach model of superconductivity: Magnetic fields parallel to the axes, J. Eur. Math. Soc., 14 (2012), 1825-1857. doi: 10.4171/JEMS/348. Google Scholar

[5]

S. AlamaL. Bronsard and E. Sandier, Minimizers of the Lawrence-Doniach functional with oblique magnetic fields, Comm. Math. Phys., 310 (2012), 237-266. doi: 10.1007/s00220-011-1399-2. Google Scholar

[6]

G. AlbertiS. Baldo and G. Orlandi, Variational convergence for functionals of Ginzburg-Landau type, Indiana Univ. Math. J., 54 (2005), 1411-1472. doi: 10.1512/iumj.2005.54.2601. Google Scholar

[7]

S. BaldoR. L. JerrardG. Orlandi and H. M. Soner, Convergence of Ginzburg-Landau functionals in three-dimensional superconductivity, Arch. Rational Mech. Anal., 205 (2012), 699-752. doi: 10.1007/s00205-012-0527-2. Google Scholar

[8]

S. BaldoR. L. JerrardG. Orlandi and H. M. Soner, Vortex density models for superconductivity and superfluidity, Comm. Math. Phys., 318 (2013), 131-171. doi: 10.1007/s00220-012-1629-2. Google Scholar

[9]

P. Bauman and Y. Ko, Analysis of solutions to the Lawrence-Doniach system for layered superconductors, SIAM J. Math. Anal., 37 (2005), 914-940. doi: 10.1137/S0036141004444597. Google Scholar

[10]

S. J. ChapmanQ. Du and M. D. Gunzburger, On the Lawrence-Doniach and anisotropic Ginzburg-Landau models for layered superconductors, SIAM J. Appl. Math., 55 (1995), 156-174. doi: 10.1137/S0036139993256837. Google Scholar

[11]

E. B. FabesM. Jr Jodeit and N. M. Rivière, Potential techniques for boundary value problems on ${C}^1$-domains, Acta Math., 141 (1978), 165-186. doi: 10.1007/BF02545747. Google Scholar

[12]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, reprint of the 1998 ed., Springer-Verlag, Berlin, 2001. Google Scholar

[13]

T. Giorgi and D. Phillips, The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model, SIAM Review, 44 (2002), 237-256. doi: 10.1137/S003614450139951. Google Scholar

[14]

Y. Iye, How anisotropic are the cuprate high Tc superconductors?, Comments Cond. Mat. Phys., 16 (1992), 89-111. Google Scholar

[15]

R. L. Jerrard and H. M. Soner, Limiting behavior of the Ginzburg-Landau functional, J. Funct. Anal., 192 (2002), 524-561. doi: 10.1006/jfan.2001.3906. Google Scholar

[16]

A. Kachmar, The ground state energy of the three-dimensional Ginzburg-Landau model in the mixed phase, J. Funct. Anal., 261 (2011), 3328-3344. doi: 10.1016/j.jfa.2011.08.002. Google Scholar

[17]

G. Peng, Convergence of the Lawrence-Doniach energy for layered superconductors with magnetic fields near $H_c_1$, SIAM J. Math. Anal., 49 (2017), 1225-1266. doi: 10.1137/16M1064398. Google Scholar

[18]

E. Sandier and S. Serfaty, On the energy of type-II superconductors in the mixed phase, Rev. Math. Phys., 12 (2000), 1219-1257. doi: 10.1142/S0129055X00000411. Google Scholar

[19]

E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity, Ann. Sci. École Norm. Sup., 33 (2000), 561–592. doi: 10.1016/S0012-9593(00)00122-1. Google Scholar

[20]

E. Sandier and S. Serfaty, The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model, SIAM J. Math. Anal., 34 (2003), 939-956. doi: 10.1137/S0036141002406084. Google Scholar

[21]

E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Progress in Nonlinear Differential Equations and Their Applications 70, Birkhäuser, Boston, 2007. Google Scholar

[22]

G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal., 59 (1984), 572-611. doi: 10.1016/0022-1236(84)90066-1. Google Scholar

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