First-Order Transitions atHc1andHc2in Type-II Superconductors

From a consideration of several different definitions of the critical value of $k$ for type-II superconductivity at temperatures below $T_c$, it is proved that the initial flux penetration is not in the form of a singly quantized isolated vortex for pure to rather dirty materials with $k$ values near $\frac{1}{\sqrt{2}}$ at temperatures below $T_c$. It is predicted that the initial flux penetration in such materials is in the form of a vortex lattice with finite spacing; the transition at the field of first flux penetration would be first order in such materials, with a discontinuity in the magnetization and a latent heat. Recent experimental evidence supporting this prediction is discussed. A similar analysis leads to the prediction that the transition from the mixed state to the normal state is first order for very dirty materials ($l\lesssim \frac{{\xi }_0}{50}$) with $k$ values near $\frac{1}{\sqrt{2}}$, in qualitative agreement with the experimental results of Ehrat and Rinderer. Finally, it is shown by both qualitative and quantitative arguments that a negative normal-superconducting wall energy is only a sufficient condition for type-II superconductivity, and not a necessary one.