Dynamics of Interfaces in Superconductors

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Abstract
The dynamics of an interface between the normal and superconducting phases under nonstationary external conditions is studied within the framework of the time-dependent Ginzburg-Landau equations of superconductivity, modified to include thermal fluctuations. An equation of motion for the interface is derived in two steps. First, the method of matched asymptotic expansions is used to derive a diffusion equation for the magnetic field in the normal phase, with nonlinear boundary conditions at the interface. These boundary conditions are a continuity equation which relates the gradient of the field at the interface to the normal velocity of the interface, and a modified Gibbs-Thomson boundary condition for the field at the interface. Second, the boundary integral method is used to integrate out the magnetic field in favor of an equation of motion for the interface. This equation of motion, which is highly nonlinear and nonlocal, exhibits a diffusive instability (the Mullins-Sekerka instability) when the superconducting phase expands into the normal phase (i.e., when the external field is reduced below the critical field). In the limit of infinite diffusion constant the equation of motion becomes local in time, and can be derived variationally from a static energy functional which includes the bulk free energy difference between the two phases, the interfacial energy, and a long range self-interaction of the interface of the Biot-Savart form. In this limit the dynamics is identical

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