The threshold temperature where type-I and type-II interchange in mesoscopic superconductors at the Bogomolnyi limit

In this work we discuss the H−T phase diagram for mesoscopic squared superconducting samples at the Bogomolnyi limit, where the Ginzburg-Landau constant κ=1/2. We calculate Hp(T), the vortex penetration field, and Hu(T) the upper critical field. Through the study of the temperature dependence on the Hp, it is possible to distinguish the region where the magnetic field penetrates into the sample, like a type-I or a type-II superconductor. It permits to determine the threshold temperature T⋆(L,H) where the phase transition from type-I to type-II occurs for some different sizes L of the mesoscopic superconducting samples. The calculation of the upper critical field Hu(T), for these samples, shows that, these two curves, Hp(T) and Hu(T), overlap at the threshold temperature mentioned above. The magnetization of the system was calculated for all sizes studied in this work, and for temperatures above and below T⋆(L,H). This study confirms the existence of the threshold temperature, T⋆(L,H), where type-I and type-II interchange in mesoscopic superconductors at the Bogomolnyi limit.