Magnetic Properties of Type-II Superconductors in the Two-Band Model

It has been suggested previously by Wong and Sung that the present discrepancies between theory and experiments on the magnetic properties of some transition metals which are intrinsic type-II superconductors (e.g., Nb) may be caused by the overlapping-band effect of $s$ and $d$ bands. We present here the numerical results of this effect on the magnetic properties. It is shown that the overlapping-band effect in the pure limit can reduce the upper critical field $H_{c2\mathrm{}}$ about 30-60% at $T$ close to temperature $T_c$, and thus enhance the over-all temperature dependence of $H_{c2\mathrm{}}$. The amount of reduction decreases as the superconductors become dirtier. Reasonable parameters which characterize the overlapping-band effect are assumed. We also compute ${\kappa }_2$, which defines the slope of $H_{c2\mathrm{}}$ when the external field is slightly less than $H_{c2\mathrm{}}$. It is found that ${\kappa }_2$ becomes smaller in comparison with the present theory as $T\to T_c$; thus a stronger temperature dependence is also associated with ${\kappa }_2$. The numerical value of this correction in ${\kappa }_2$, however, is very sensitive to the parameters used.