Large Broadening of the Superconducting Transition by Fluctuations in a 3D Metal at High Magnetic Fields: The MgB$_{2}$ case

It is shown that the transition to the low temperature superconducting state in a 3D metal at high magnetic field is smeared dramatically by thermal fluctuation of the superconducting order parameter. The resulting superconducting-to-normal crossover occurs in a vortex liquid state which is extended well below the mean-field H c 2 . Application to MgB 2 yields good quantitative agreement with recently reported data of dHvA oscillation in the superconducting state.