Critical current in thin flat superconductors with Bean-Livingston and geometrical barriers

Dependence of the critical current $I_c$ on the applied magnetic field $H_a$ is theoretically studied for a thin superconducting strip of a rectangular cross section, taking an interplay between the Bean-Livingston and the geometric barriers in the sample into account. It is assumed that bulk vortex pinning is negligible, and the London penetration depth $\lambda$ is essentially less than the thickness $d$ of the strip. To investigate the effect of these barriers on $I_c$ rigorously, a two-dimensional distribution of the current over the cross section of the sample is derived, using the approach based on the methods of conformal mappings. With this distribution, the dependence $I_c(H_a)$ is calculated for the fields $H_a$ not exceeding the lower critical field. This calculation reveals that the following two situations are possible: i) The critical current $I_c(H_a)$ is determined by the Bean-Livingston barrier in the corners of the strip. ii) The geometrical barrier prevails at low $H_a$, but with increasing magnetic field, the Bean-Livingston barrier begins to dominate. The realization of one or the other of these two situations is determined by the ratio $\lambda/d$.