Small Superconducting Grain in the Canonical Ensemble (original) (raw)
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Physical Review B, 2003
Unlike bulk superconductivity, where one energy scale, the energy gap, characterizes pairing correlations, we show that in small superconducting grains there exist two different such quantities. The first characterizes cumulative properties of the grain, such as the condensation energy, and the second single-particle properties. To describe these two energy scales, we define two corresponding pairing parameters, and show that although both reduce to the bulk gap for large grains, this occurs at different size scales.
Superconducting correlations in ultra-small metallic grains
Europhysics Letters (EPL), 2000
To describe the crossover from the bulk BCS superconductivity to a fluctuationdominated regime in ultrasmall metallic grains, new order parameters and correlation functions, such as "parity gap" and "pair-mixing correlation function", have been recently introduced. In this paper, we discuss the small-grain behaviour of the Penrose-Onsager-Yang off-diagonal long-range order (ODLRO) parameter in a pseudo-spin representation. Relations between the ODLRO parameter and those mentioned above are established through analytical and numerical calculations.
Exact study of the effect of level statistics in ultrasmall superconducting grains
Physical Review B, 2000
The reduced BCS model that is commonly used for ultrasmall superconducting grains has an exact solution worked out long ago by Richardson in the context of nuclear physics. We use it to check the quality of previous treatments of this model, and to investigate the effect of level statistics on pairing correlations. We find that the ground state energies are on average somewhat lower for systems with non-uniform than uniform level spacings, but both have an equally smooth crossover from the bulk to the few-electron regime. In the latter, statistical fluctuations in ground state energies strongly depend on the grain's electron number parity.
Strong-coupling expansion for the pairing Hamiltonian for small superconducting metallic grains
Physical Review B, 2003
The paper is devoted to the study of the effects due to superconducting pairing in small metallic grains. We explicitly determine the low-energy spectrum of the problem at strong superconducting coupling and in the limit of large Thouless conductance. We start with the strong-coupling limit and develop a systematic expansion in powers of the inverse coupling constant for the many-particle spectrum of the system. The strongcoupling expansion is based on the formal exact solution of the Richardson model and converges for realistic values of the coupling constant. We use this expansion to study the low-energy excitations of the system, in particular energy and spin gaps in the many-body spectrum.
Journal of Physics: Condensed Matter, 2012
Superconducting correlations in an isolated metallic grain are governed by the interplay between two energy scales: the mean level spacing δ and the bulk pairing gap 0 , which are strongly influenced by the position of the chemical potential with respect to the closest single-electron level. In turn superconducting correlations affect the position of the chemical potential. Within the parity projected BCS model we investigate the probability distribution of the chemical potential in a superconducting grain with randomly distributed single-electron levels. Taking into account statistical fluctuations of the chemical potential due to the pairing interaction, we find that such fluctuations have a significant impact on the critical level spacing δ c at which the superconducting correlations cease: the critical ratio δ c / 0 at which superconductivity disappears is found to be increased.
Physical Review B - PHYS REV B, 2003
We investigate the effects of gap fluctuations on the pair-transfer correlation and spectral functions in nanoscopic superconducting grains at finite temperature, by means of the correlated static path approximation (CSPA). The present approach is able to provide a reliable description of the lowest energy weighted moments of the spectral function of small samples in critical regions, improving both the random-phase approximation (RPA) and the SPA predictions. The results confirm the persistence of pairing effects in the spectral function beyond the BCS critical temperatures and sizes, which is visible through the enhancement of the strength at low energies and the concomitant decrease in the normalized first energy moment. The role played by the zero and the imaginary quasiparticle RPA energies present in the CSPA is also discussed.
Thermodynamic properties of a small superconducting grain
Physical Review B, 2001
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 ⌬ and the small regime, where ⌬տd, 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 and is proportional to dN 2 ϭ 2 D. We find that also in a large regime with ⌬Ͼd, 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 ⌬Ͼͱd 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 ⌬.
Temperature and magnetic field dependence of superconductivity in nanoscopic metallic grains
Solid State Communications, 2001
We study pairing correlations in ultrasmall superconductor in the nanoscopic limit by means of a toy model where electrons are con®ned in a single, multiply degenerate energy level. We solve the model exactly to investigate the temperature and magnetic ®eld dependence of number parity effect (dependence of ground state energy on evenness or oddness of the number of electrons). We ®nd a different parity effect parameter to critical temperature ratio (.4 rather than 3.5) which turns out to be consistent with exact solution of the BCS gap equation for our model. This suggests the equivalence between the parity effect parameter and the superconducting gap. We also ®nd that magnetic ®eld is suppressed as temperature increases. q
Parity-Affected Superconductivity in Ultrasmall Metallic Grains
Physical Review Letters, 1996
We investigate the breakdown of BCS superconductivity in ultrasmall metallic grains as a function of particle size (characterized by the mean spacing d between discrete electronic eigenstates), and the parity (P = even/odd) of the number of electrons on the island. Assuming equally spaced levels, we solve the parity-dependent BCS gap equation for the order parameter ∆P (d, T). Both the T = 0 critical level spacing dc,P and the critical temperature Tc,P (d) at which ∆P = 0 are parity dependent, and both are so much smaller in the odd than the even case that these differences should be measurable in current experiments.
Pair correlations and the survival of superconductivity in and around a superconducting impurity
Physical Review B, 2007
The problem of the survival of superconductivity in a small super-conducting grain placed in a metal substrate is addressed. For this aim the pair correlations and super-conducting gap around and inside a negative-U impurity in one and two dimensions is calculated, in a discrete tight-binding model and a continuous model. Using a mean-field decomposition, it is shown that finite pairing in the grain develops when the system has a degeneracy between successive number of electron pairs, and thus may oscillate as a function of the chemical potential. For finite pairing in the island, pair correlations in the normal part exhibit a cross-over from being long-ranged to exponentially decaying, depending on the strength of interaction in the grain. It is shown analytically that there is a minimal island size under-which pairing vanishes which is different than that given by Anderson's criterion, and that it scales as a power-law with island size, rather then exponentially as in isolated grains.