Analytical model of the response of a superconducting film to line currents

We theoretically investigate the response of a superconducting film to line currents flowing in linear wires placed above the film, and we present analytic expressions for the magnetic-field and current distributions based on the critical state model. The behavior of the superconducting film is characterized by the sheet-current density $K_z$, whose magnitude cannot exceed the critical value $j_cd$, where $j_c$ is the critical current density and $d$ is the thickness of the film. When the transport current $I_0$ flowing in the wire is small enough, $|K_z|$ is smaller than $j_cd$ and the magnetic field is shielded below the film. When $I_0$ exceeds a threshold value $I_{c0}\propto j_cd$, on the other hand, $|K_z|$ reaches $j_cd$ and the magnetic field penetrates below the film. We also calculate the ac response of the film when an ac transport current flows in the linear wires.