Shapes of flux domains in the intermediate state of type-I superconductors

In the intermediate state of a thin type-I superconductor magnetic flux penetrates in a disordered set of highly branched and fingered macroscopic domains. To understand these shapes, we study in detail a recently proposed "current-loop" model [R. E. Goldstein, D. P. Jackson, and A. T. Dorsey, Phys. Rev. Lett. 76, 3818 (1996)] that models the intermediate state as a collection of tense current ribbons flowing along the superconducting-normal interfaces and subject to the constraint of global flux conservation. The validity of this model is tested through a detailed reanalysis of Landau's original conformal mapping treatment of the laminar state, in which the superconductor-normal interfaces are hared within the slob, and of a closely related straight-lamina model. A simplified dynamical model is described that elucidates the nature of possible shape instabilities of flux stripes and stripe arrays, and numerical studies of the highly nonlinear regime of those instabilities demonstrate patterns like those seen experimentally. Of particular interest is the buckling instability commonly seen in the intermediate state. The free-boundary approach further allows for a. calculation of the elastic properties of the laminar state, which closely resembles that of smectic liquid crystals. We suggest several experiments to explore flux domain shape instabilities, including an Eckhaus instability induced by changing the out-of-plane magnetic field and an analog of the Helfrich-Hurault instability of smectics induced by an in-plane field.