Thermodynamic properties of a small superconducting grain
Abstract
The reduced BCS Hamiltonian for a metallic grain with a finite number of electrons is considered. The crossover between the ultrasmall regime, in which the level spacing d is larger than the bulk superconducting gap Δ and the small regime, where Δ>~d, 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, where N is the number of interacting electron pairs. The condensation energy in this regime is perturbative in the coupling constant λ and is proportional to dNλ^{2}=λ^{2}ω_{D}. We find that also in a large regime with Δ>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 be well approximated by the BCS result is found to be roughly Δ>dω_{D}. We show how the condensation energy can, in principle, be extracted from a measurement of the spin magnetization curve and find a reentrant susceptibility at zero temperature as a function of magnetic field, which can serve as a sensitive probe for the existence of superconducting correlations in ultrasmall grains. Numerical results are presented, which suggest that in the large N limit the 1/N correction to the BCS result for the condensation energy is larger than Δ.
 Publication:

Physical Review B
 Pub Date:
 June 2001
 DOI:
 10.1103/PhysRevB.63.214518
 arXiv:
 arXiv:condmat/0012160
 Bibcode:
 2001PhRvB..63u4518S
 Keywords:

 74.20.Fg;
 74.80.Bj;
 74.25.Ha;
 74.25.Bt;
 BCS theory and its development;
 Magnetic properties;
 Thermodynamic properties;
 Condensed Matter  Superconductivity
 EPrint:
 17 pages, 7 figures, Submitted to Phys. Rev. B