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 $dN{\lambda }^2={\lambda }^2{\omega }_D$. 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 $\Delta$.

17 pages, 7 figures, Submitted to Phys. Rev. B

Cited by in corpus (40)

• Exactly solvable Richardson-Gaudin models for many-body quantum systems
• Electron spin dynamics in quantum dots and related nanostructures due to hyperfine interaction with nuclei
• Integrable model for interacting electrons in metallic grains
• Large N limit of the exactly solvable BCS model: analytics versus numerics
• Exact mesoscopic correlation functions of the pairing model
• Strong Coupling Expansion for the Pairing Hamiltonian
• BCS theory for finite size superconductors
• Finite Size Corrections for the Pairing Hamiltonian
• Accuracy of BCS-based approximations for pairing in small Fermi systems
• Integrable models for confined fermions: applications to metallic grains
• Exact correlation functions of the BCS model in the canonical ensemble
• BCS superconductivity in metallic nanograins: Finite-size corrections, low energy excitations, and robustness of shell effects
• Quantifying ground state entanglement of the BCS model
• Electrostatic analogy for integrable pairing force Hamiltonians
• An exactly solvable BCS-Hubbard Model in arbitrary dimensions
• Shape of nanosize superconducting grains: Does it influence pairing characteristics?
• The magnetic response of disordered metallic rings: the large contribution of the far levels
• Dynamical correlation functions of the mesoscopic pairing model
• Magnetic susceptibility of ultra-small superconductor grains
• The n_s - T_c correlations in granular superconductors
• Projected Quasi-particle Perturbation theory
• Mesoscopic fluctuations in superconducting dots at finite temperatures
• Thermodynamics of ultrasmall metallic grains in the presence of pairing and exchange correlations: Mesoscopic fluctuations
• Josephson effect between superconducting nanograins with discrete energy levels
• Integrability and exact solution for coupled BCS systems associated with the $su(4)$ Lie algebra
• Spin magnetization of small metallic grains
• Pairing Correlations in Finite Systems: From the weak to the strong fluctuations regime
• Bethe Ansatz approach to the pairing fluctuations in the mesoscopic regime
• Two pairing parameters in superconducting grains
• Coboson formalism for Cooper pairs used to derive Richardson's equations
• On quantum phase crossovers in finite systems
• Exact solution for the inhomogeneous Dicke model in the canonical ensemble: thermodynamical limit and finite-size corrections
• Pairing correlations and eigenvalues of two-body density matrix in atomic nuclei
• Magnetic extrema in electronic susceptibility and heat capacity of mesoscopic systems
• Electron-hole symmetry and solutions of Richardson pairing model
• The coexistence of superconductivity and ferromagnetism in nano-scale metallic grains
• Well-defined quasiparticles in interacting metallic grains
• Applicability of Bardeen-Cooper-Schrieffer theory to small-sized superconductors: role of Cooper-pair binding energy
• The many levels pairing Hamiltonian for two pairs
• Superconducting fluctuations in small grains - the Universal Hamiltonian and the reduced BCS model