The critical current density in type-II superconducting bulk materials
  The critical current density Jc in type-II conventional and high-Tc superconducting bulk materials is investigated based on the quantum theory for the vortex dynamics. It is shown that for a constant magnetic field, the critical current density Jc decreases weakly with increasing temperature when T<T_dp(depinning temperature); when T_dp< T <T_f (boundary fluctuation temperature,Jc is power-law-decaying,and when T>T_f,Jc decays exponentially; while for a constant temperature, Jc first decreases, then increases after reaching a maximum, and finally decreases again as the magnetic field increases. These results are in agreement with the experiments. 
Ming Ju Chou and Wei Yeu Chen
cond-mat.supr-con