Simple vortex states in films of type-I Ginzburg-Landau superconductor

Sufficiently thin films of type-I superconductor in a perpendicular magnetic field exhibit a triangular vortex lattice while thick films develop an intermediate state. To elucidate what happens between these two regimes, precise numerical calculations have been made within Ginzburg-Landau theory at κ=0.5 and 0.25 for a variety of vortex lattice structures with one flux quantum per unit cell. The phase diagram in the space of mean induction and film thickness includes a narrow wedge in which a square lattice is stable, surrounded by the domain of stability of the triangular lattice at thinner films/lower fields and, on the other side, rectangular lattices with continuously varying aspect ratio. The vortex lattice has an anomalously small shear modulus within and close to the square lattice phase. Solutions of the Ginzburg-Landau equations have been obtained by similar calculations for bulk systems and thin films with one vortex but two flux quanta per square or triangular unit cell. Primitive lattices of double-fluxoid vortices are thermodynamically unstable in bulk in both type-I and type-II superconductors, as expected. In type-I films these double-fluxoid lattices do not pre-empt the single-fluxoid lattice structures.