The Mixed State of Thin Superconducting Films in Perpendicular Fields

The behavior of ideal thin superconducting films in perpendicular magnetic fields is studied in detail and related to that of bulk type-II superconductors. A macroscopic analysis based only on the demagnetizing factors yields the dominant effects of sample geometry on the reversible magnetization curve. The same features are also derived from Pearl's generalization of Abrikosov's microscopic model, which predicts a long-range interaction between quantized flux lines in a thin film. Comparison of the macroscopic and microscopic arguments clarifies some inaccuracies in the work of Pearl and of Maki. The dependence of critical magnetic fields on film thickness is discussed for different values of the Ginzburg—Landau parameter $\kappa$. A hydrodynamic calculation demonstrates that a triangular vortex lattice is stable against small perturbations in the long-wavelength limit ($q{n}^{\frac{-1\mathrm{}}{2}}\ll 1$); for $n^{\frac{1}{2}}\Lambda \gg 1$, the corresponding dispersion relation is $\omega =\frac{1}{4}\left(\frac{\mathrm{eB}}{\mathrm{mc}}\right)q^{\frac{3}{2}}{\Lambda }^{\frac{1}{2}}{\left(n\pi \right)}^{\frac{-1\mathrm{}}{2}}$, where $n$ is the vortex density, $\Lambda \equiv \frac{2{\lambda }^2}{d}$ is the "effective penetration depth," $\lambda$ is the actual penetration depth, and $d(\ll \lambda )$ is the film thickness. This conclusion disagrees with Pearl's conjecture based on elasticity theory; the long-range interaction precludes the use of elasticity theory, as is seen from the difference between the calculated dispersion relation ($\omega \propto q^{\frac{3}{2}}$) and that predicted for elastic modes ($\omega \propto q^2$). The dynamics of vortex systems is contrasted with the Newtonian dynamics of point masses. In practice, thin films exhibit highly irreversible behavior, and no detailed comparison between theory and experiment is attempted.