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

Patricia Bauman; Guanying Peng

doi:10.3934/dcdsb.2019112

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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 T_c 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 T_c 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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