Thermodynamic properties of a small superconducting grain - art. no. 214518

The reduced BCS Hamiltonian for a metallic grain with a finite number of el ectrons is considered. The crossover between the ultrasmall regime, in whic h the level spacing d is larger than the bulk superconducting gap Delta and the small regime, where Delta greater than or similar tod, is investigated analytically and numerically. The condensation energy, spin magnetization, and tunneling peak spectrum are calculated analytically in the ultrasmall regime, using an approximation controlled by 1/ln N as a small parameter, w here N is the number of interacting electron pairs. The condensation energy in this regime is perturbative in the coupling constant lambda and is prop ortional to dN lambda (2) = lambda (2)omega (D). We find that also in a lar ge regime with Delta > d, in which pairing correlations are already rather well developed, the perturbative part of the condensation energy is larger than the singular, BCS part. The condition for the condensation energy to b e well approximated by the BCS result-is found to be roughly Delta rootd om ega (D). We show how the condensation energy can, in principle, be extracte d from a measurement of the spin magnetization curve and find a reentrant s usceptibility at zero temperature as a function of magnetic field, which ca n serve as a sensitive probe for the existence of superconducting correlati ons in ultrasmall grains. Numerical results are presented, which suggest th at in the large N limit the 1/N correction to the BCS result for the conden sation energy is larger than Delta.