The thermodynamic properties of a mesoscopic-size, simply connected cylindrical normal metal in good metallic contact with superconducting banks are studied theoretically. It is commonly accepted that if the superconductor thickness is quite small (of the order of the coherence length), as is assumed to be the case here, a vector potential field, whose value can be varied, exists inside the normal layer. It is further assumed that the quasiparticles with energy E<Δ (2Δ is the superconducting gap) move ballistically through the normal metal and undergo Andreev scattering caused by the off-diagonal potential of the superconductor. An equation is obtained within the multidimensional quasiclassical method which permits us to determine the spectrum of the Andreev levels and to calculate the density of states of the system in question. It is shown that the Andreev levels shift as the trapped flux Φ changes inside the normal conductor. At a certain flux value they coincide with the Fermi level. A resonance spike in the density of states ν(E) appears in this case, since near E=0 there is strong degeneracy of the quasiparticle states in respect to the quantum number q characterizing their motion along the cylinder axis. As a result, a macroscopic number of q states contribute to the amplitude of the effect. As the flux is increased, the density of states v(E) behaves as a stepwise function of Φ. The distance between the steps is equal to the superconducting flux quantum hc/2e.
Quantum states of a superconductor–insulator–normal metal–insulator–superconductor sandwich (the SINIS structure) are investigated on the basis of the Bogoliubov–de Gennes equations. The dispersion equation is obtained for the quasiparticle spectrum for energies E<Δ (Δ is the energy gap in the superconductor) taking into account the Andreev scattering as well as conventional electron reflection at the interfaces of the SINIS structure. The spectrum makes it possible to calculate the Josephson current in the system. The transparency coefficient of the system for electrons with a continuous energy spectrum is calculated, and quasi-local states (“resonance levels” of transparency) are determined for the structure under investigation.
The coherent phenomena in mesoscopic cylindrical normal metal (N) -superconductor (S) structures have been investigated theoretically. The magnetic moment (persistent current) of such a structure has been calculated numerically and (approximately) analytically. It is shown that the current in the N-layer corresponding to the free energy minimum is always diamagnetic. As the field increases, the magnetic moment (current) exhibits jumps at certain values of the trapped magnetic flux and the NS structure changes to a state with smaller absolute value of the diamagnetic moment. This occurs when the persistent current is unable to screen the external field. The magnetic moment increase stepwise and the system changes into a new stable state. The magnetic field penetrates into a larger volume of the N-layer. The new state has smaller absolute value of the diamagnetic moment. Experimentally, this is interpreted as the presence of a paramagnetic addition in the system (paramagnetic reentrant effect). The results obtained are in qualitative agreement with the experiments conducted by P. Visani, A. C.Mota, and A. Pollini, Phys. Rev. Lett. 65, 1514Lett. 65, (1990.
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