Mixed State of Type-I Superconducting Films in a Perpendicular Magnetic Field

Thin films of type-I superconductors are shown to exist in a variety of quite distinct mixed states, depending on their thicknesses. The most interesting of these states consist of hexagonal arrays of vortices of larger than unit quantum number. Their general character falls between that of the ordinary mixed state, a triangular array of unit vortices which occurs for sufficiently thin films, and the intermediate states consisting of islands of superconducting phase in a matrix of normal phase which occurs for sufficiently thick films. The theory is developed in Abrikosov's high-field approximation, which gives solutions of the Ginzburg-Landau equations that are exact in the limit as the applied field approaches the second critical field ${\mathrm{H}}_{c2\mathrm{}}$. Two critical thicknesses are found and determined as functions of the Ginzburg-Landau parameter $\kappa$. The first and smaller critical thickness is the maximum film thickness for which the ordinary mixed state will exist near ${\mathrm{H}}_{c2\mathrm{}}$. The second critical thickness is the maximum for which a second-order field transition occurs at ${\mathrm{H}}_{c2\mathrm{}}$. The new types of mixed state are stable for values of film thickness intermediate between these two.