The theory of the reentrant effect in susceptibility of cylindrical   mesoscopic samples
  A theory has been developed to explain the anomalous behavior of the magnetic susceptibility of a normal metal-superconductor ($NS$) structure in weak magnetic fields at millikelvin temperatures. The effect was discovered experimentally by A.C. Mota et al \cite{10}. In cylindrical superconducting samples covered with a thin normal pure metal layer, the susceptibility exhibited a reentrant effect: it started to increase unexpectedly when the temperature lowered below 100 mK. The effect was observed in mesoscopic $NS$ structures when the $N$ and $S$ metals were in good electric contact. The theory proposed is essentially based on the properties of the Andreev levels in the normal metal. When the magnetic field (or temperature) changes, each of the Andreev levels coincides from time to time with the chemical potential of the metal. As a result, the state of the $NS$ structure experiences strong degeneracy, and the quasiparticle density of states exhibits resonance spikes. This generates a large paramagnetic contribution to the susceptibility, which adds up to the diamagnetic contribution thus leading to the reentrant effect. The explanation proposed was obtained within the model of free electrons. The theory provides a good description for experimental results [10]. 
G.A. Gogadze