The Landau-Ginzburg free-energy expansion of the superconducting order parameter in the presence of a magnetic vector potential has been used as a basis for the analysis of magnetic-field penetration in superconductors. Several specific cases have been examined in one and two dimensions in order to solve the complicated system of coupled nonlinear partial differential equations that describe interactions between the magnetic field and superconducting charge density. Exact solutions are found at T=${\mathit{T}}_{\mathit{c}}$, while accurate series expansions are used close to it. Oscillatory damped profiles of the magnetic-field penetration are found. In addition, periodic patterns of magnetic induction and a phase-shifted superconducting charge density are obtained. Other approximate methods were used to examine the behavior of the superconducting system for arbitrary temperatures below the critical temperature. We have also found two types of two-dimensional solutions, i.e., vortices, which exist below ${\mathit{T}}_{\mathit{c}}$, and spirals, which appear to exist at or very near to T=${\mathit{T}}_{\mathit{c}}$.

• Received 3 May 1991

DOI:https://doi.org/10.1103/PhysRevB.44.11933