Superconductivity in mesoscopic metal particles

Recently, it has been possible to construct single-electron transistors to study electronic properties, including superconductivity, in metallic grains of nanometer size. Among several theoretical results are suppression of superconductivity with decreasing grain size and parity effect (i.e., dependence on the parity of the number of electrons on the grain). We study how these results are affected by degeneracy of energy levels. In addition to the time-reversal symmetry, for certain energy spectra and more generally for lattice symmetries, energy levels degenerate strongly near the Fermi energy. For a parabolic dispersion, degeneracy d is of the order of k(F)L, whereas the typical distance between the levels is of the order of epsilon (F)/(k(F)L)(2), where k(F) and epsilon (F) are the Fermi wave-vector and energy, respectively, and L is the particle size. First, using an exact solution method for BCS Hamiltonian with finite number of energy levels, for the well-studied nondegenerate case we find a new feature. In that case, parity effect exhibits a minimum instead of a monotonic behavior. For d-fold degenerate states, we find that the ratio of two successive parity effect parameters Delta (p) is nearly 1+1/d. Our numeric solutions for the exact ground-state energy of negative U Hubbard model on a cubic cluster also give very similar results. Hence we conclude that parity effect is a general property of small Fermi systems with attractive interaction, and it is closely related to degeneracy of energy levels.