Abstract
Electrically conducting films in a time-varying transverse applied magnetic field are considered. Their behavior is strongly influenced by the self-field of the induced currents, making the electrodynamics nonlocal, and consequently difficult to analyze both numerically and analytically. We present a formalism which allows many phenomena related to superconducting and Ohmic films to be modeled and analyzed. The formalism is based on the Maxwell equations and a material current–voltage characteristics, linear for normal metals and nonlinear for superconductors, plus a careful account of the boundary conditions. For Ohmic films, we consider the response to a delta function source-field turned on instantly. As one of few problems in nonlocal electrodynamics, this has an analytical solution, which we obtain in both Fourier and real space. Next, the dynamical behavior of a square superconductor film during ramping up of the field, and subsequently returning to zero, is treated numerically. Then, this remanent state is used as initial condition for triggering thermomagnetic avalanches. The avalanches tend to invade the central part where the density of trapped flux is largest, forming dendritic patterns in excellent agreement with magneto-optical images. Detailed profiles of current and flux density are presented and discussed. Finally, the formalism is extended to multiply connected samples, and numerical results for a patterned superconducting film, a ring with a square lattice of antidots, are presented and discussed.
Highlights
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The macroscopic electrodynamics of thin films, either superconducting or Ohmic, in transverse applied field can be modeled by the Maxwell equations
The formalism is capable of handling a wide range of physical systems, where the material-specific properties are introduced as an E–J relation, which is linear for Ohmic conductors, nonlinear for superconductors
Summary
The current density in the film can be expressed as j = J(x, y)δ(z),. The total magnetic moment of the film can be expressed as 1 mz = 2 r × j(r) d3r = zg dx dy. Fourier transforms along the Cartesian axes give iky Hz[3] − ikz Hy[3] = iky g[2],. Conservation of magnetic flux, ∇ · H = 0, yields ikx Hx[3] + iky Hy[3] + ikz Hz[3] = 0, so that. K2 g[2], kz2 + k2 and inverse Fourier transform in z direction results in the final expression. Where F and F−1 is forward and inverse Fourier transform, respectively. For films having a finite area, they are good approximations for short wavelengths [1]
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