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, $\Delta$, and the small regime, where $\Delta \gtrsim d$, 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 small parameter, where $N$ is the number of interacting electron pairs. The condensation energy in this regime is perturbative in the coupling constant $\lambda$, and is proportional to $d N \lambda^2 = \lambda^2 \omega_D$. We find that also in a large 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 be well approximated by the BCS result is found to be roughly $\Delta > \sqrt{d \omega_D}$. We show how the condensation energy can, in principle, be extracted from a measurement of the spin magnetization curve, and find a re-entrant 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 $\Delta$.Comment: 17 pages, 7 figures, Submitted to Phys. Rev.

Bardeen-Cooper-Schrieffer Theory of Finite-Size Superconducting Metallic Grains

“…DOI: 10.1103/PhysRevLett.100.187001 PACS numbers: 74.20.Fg, 05.45.Mt, 74.78.Na Since experiments by Ralph, Black, and Tinkham [1] on Al nanograins in the mid-1990s, there has been considerable interest in the theory of ultrasmall superconductors (see [2,3] for earlier studies). In particular, finite-size corrections to the predictions of the Bardeen-CooperSchrieffer (BCS) theory for bulk superconductors [4] have been studied [5][6][7][8][9][10] within the exactly solvable Richardson model [11]. Pairing in specific potentials, such as a harmonic oscillator potential [12] and a rectangular box, [13,14] and mesoscopic fluctuations of the energy gap [15,16] have been explored as well.…”

Current-current correlations in hybrid superconducting and normal-metal multiterminal structures

“…Current noise measurement revealed a powerful probe for electronic systems [7]. In particular, cross-correlation in multiterminal superconducting hybrid structures has deserved some attention since it was first anticipated [8,9] and then predicted [10,11,12,13,14,15,16,17] to show a sign change with respect to normal metallic structures. A simplified explanation of this effect is the following.…”

“…By contrast, electrons from a fermionic source due to Pauli principle arrive one by one in each channel and they are transmitted either in one or the other lead, leading to a negative correlation. Most calculations consider Y-shaped structures attached to a superconducting lead through a single junction [10,11,12,13,15,16,17]. This geometry is the standard one for interference in optical experiments, but it leads to a variety of elementary processes for charge transfer in electronic samples.…”

“…One can then safely discard voltage and ε dependence of F in Eq. (13). A similar procedure can be applied to S EC as well.…”

Mesoscopic superconductivity in ultrasmall metallic grains

“…Here we review signatures of this competition in thermodynamic properties of the nanoparticle such as the heat capacity and spin susceptibility [13,12,14]. Numberparity effects, induced by pairing correlations [15,16,17,18,19,20,21], are modified by the exchange interaction in a way that is qualitatively different between the fluctuation-dominated and BCS regimes. This is discussed below, and more details can be found in Refs.…”