Exact solutions of the Lawrence-Doniach model for layered superconductors

Abstract

We solve the problem of exact minimization of the Lawrence-Doniach (LD) free-energy functional in parallel magnetic fields. We consider both the infinite in the layering direction case (the infinite LD model) and the finite one (the finite LD model). We prove that, contrary to a prevailing view, the infinite LD model does not admit solutions in the form of isolated Josephson vortices. For the infinite LD model, we derive a closed, self-consistent system of mean-field equations involving only two variables. Exact solutions to these equations prove simultaneous penetration of Josephson vortices into all the barriers, accompanied by oscillations and jumps of the magnetization, and yield a completely new expression for the lower critical field. Moreover, the obtained equations allow us to make self-consistent refinements on such well-known results as the Meissner state, Fraunhofer oscillations of the critical Josephson current, the upper critical field, and the vortex solution of Theodorakis [S. Theodorakis, Phys. Rev. B 42, 10 172 (1990)]. Our consideration of the finite LD model illuminates the role of the boundary effect. In contrast to the infinite case, an explicit analytical solution to the Maxwell equations of the finite case does not preclude the existence of localized Josephson vortex configurations. By the use of this solution, we obtain a self-consistent description of the Meissner state. Finally, we discuss some theoretical and experimental implications.