We consider periodic minimizers of the Lawrence–Doniach functional, which models highly anisotropic superconductors with layered structure, in the simultaneous limit as the layer thickness tends to zero and the Ginzburg–Landau parameter tends to infinity. In particular, we consider the properties of minimizers when the system is subjected to an external magnetic field applied either tangentially or normally to the superconducting planes. For normally applied fields, our results show that the resulting “pancake” vortices will be vertically aligned. In horizontal fields we show that there are two-parameter regimes in which minimizers exhibit very different characteristics. The low-field regime resembles the Ginzburg–Landau model, while the high-field limit gives a “transparent state” described in the physical literature. To obtain our results we derive sharp matching upper and lower bounds on the global minimizers of the energy.
Cite this article
Stan Alama, Lia Bronsard, Etienne Sandier, On the Lawrence–Doniach model of superconductivity: magnetic fields parallel to the axes. J. Eur. Math. Soc. 14 (2012), no. 6, pp. 1825–1857