Effect of Orbital Quantization on the Critical Field of Type-II Superconductors

We discuss the effect of the discreteness of the Landau levels on the upper critical field of a type-II superconductor. At $T=0\mathrm{}°$K an ideal superconductor will remain superconducting in an arbitrarily large magnetic field. However, the high-field state is destroyed by a minute amount of impurity scattering or by a small misalignment of up- and down-spin Landau levels. The residual effects are (a) a small shift in the low-temperature critical field [$\frac{\Delta H_{c2\mathrm{}}}{H_{c2\mathrm{}}}\approx {\left(k_F{\xi }_0\right)}^{-1\mathrm{}}$] and (b) oscillatory structure in the transition temperature $T_{c2\mathrm{}}\mathrm{}\left(H\right)\mathrm{}$ for temperatures $T\lesssim \frac{\hslash {\omega }_c}{2{\pi }^{2}k}$.