Thermodynamic properties of a small superconducting grain

M. Schechter, Y. Imry, Y. Levinson, J. Delft

Research output: Contribution to journalArticlepeer-review

61 Scopus citations


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 1/1n 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 dNλ2 = λ2ωD. 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> √dωD. 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.

Original languageEnglish
Article number214518
Pages (from-to)2145181-21451816
Number of pages19306636
JournalPhysical Review B - Condensed Matter and Materials Physics
Issue number21
StatePublished - 1 Jan 2001
Externally publishedYes

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics


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