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, that is, 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 are strongly degenerate near the Fermi energy. For a parabolic dispersion, degeneracy d is of the order of kF L, whereas the typical distance between the levels is of the order of ∈F/(kFL)2 where kF and ∈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, we find a new feature for the well studied nondegenerate case. 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 Δp is nearly 1 + 1 /d. Our numerical 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.