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

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

Patricia Bauman 1, and  Guanying Peng 2,,

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) $.

Keywords: Superconductor, Lawrence-Doniach energy, layered superconductor, Ginzburg-Landau energy, minimizer.
Mathematics Subject Classification: Primary: 35J60, 35J65, 35Q40.
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. Alama, A. 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. Alama, L. 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. Alama, L. 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. Alama, L. 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. Alama, L. 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. Alberti, S. 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. Baldo, R. L. Jerrard, G. 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. Baldo, R. L. Jerrard, G. 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. Chapman, Q. 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. Fabes, M. 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. Alama, A. 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. Alama, L. 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. Alama, L. 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. Alama, L. 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. Alama, L. 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. Alberti, S. 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. Baldo, R. L. Jerrard, G. 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. Baldo, R. L. Jerrard, G. 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. Chapman, Q. 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. Fabes, M. 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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