Parity Effect in Ground State Energies of Ultrasmall Superconducting Grains

Abstract

We study the superconductivity in small grains in the regime when the quantum level spacing δε is comparable to the gap ∆. As δε is increased, the system crosses over from superconducting to normal state. This crossover is studied by calculating the dependence of the ground state energy of a grain on the parity of the number of electrons. The states with odd numbers of particles carry an additional energy ∆P , which shows non-monotonic dependence on δε. Our predictions can be tested experimentally by studying the parity-induced alternation of Coulomb blockade peak spacings in grains of different sizes.The standard BCS theory [1] gives a good description of the phenomenon of superconductivity in large samples. However, it was noticed by Anderson [2] in 1959 that as the size of a superconductor becomes smaller, and the quantum level spacing in the sample δε approaches the superconducting gap ∆, the BCS theory fails. The interest to the superconductivity in such ultrasmall grains was renewed by recent experiments by Ralph, Black and Tinkham [3,4], who fabricated and studied nanometerscale aluminum grains. In qualitative agreement with the prediction [2], they demonstrated [4] the existence of superconducting gap in relatively large grains, with estimated level spacings δε ≈ 0.02 and 0.08 meV smaller than the superconducting gap ∆ ≈ 0.31 meV, whereas no signs of superconductivity were observed [3] in smaller grains, δε ≈ 0.7 meV. These experiments raise a theoretical question about the nature of the crossover from superconducting to normal state in ultrasmall particles with level spacings δε ∼ ∆.This problem was addressed in two recent theoretical papers. J. von Delft et al.[5] explored the BCS gap equation in a finite-size system with equidistant discrete energy levels, and found that as the level spacing is increased, the superconducting gap of the grain vanishes at a certain critical value of δε, which is of order ∆ and depends on the parity of the total number of electrons in the grain. Smith and Ambegaokar [6] extended the treatment of Ref.[5] to take into account Wigner-Dyson fluctuations of the energy levels in the grain.It is worth noting that the theories [5,6] treat the superconductivity in small grains within the selfconsistent mean field approximation for the superconducting order parameter. Although this approximation works well for large systems, one should expect the quantum fluctuations of the order parameter to grow when the level spacing δε reaches ∆. In this paper we present a theory of superconductivity in ultrasmall grains which includes the effects of quantum fluctuations of the order parameter.We show that the corrections to the mean field results which are small in large grains, δε ≪ ∆, become important in the opposite limit, δε ≫ ∆.The superconducting gap ∆ studied in Refs. [5,6] is not well defined in the presence of quantum fluctuations. Therefore, we must first identify an observable physical quantity which characterizes the superconducting properties of small grains. The most convenient s...

Spectroscopy of discrete energy levels in ultrasmall metallic grains

“…RBT's new experiments stimulated a number of theoretical attempts to quantitatively describe the crossover from the bulk limit d ≪∆, where superconductivity is well-developed, to the fluctuation-dominated regime of d ≫∆, where pairing correlations survive only in the form of weak fluctuations. Describing this crossover constituted a conceptual challenge, since the standard grand-canonical mean-field BCS treatment of pairing correlations [21][22][23][24][25][26][27][28] breaks down for d ∆ . This challenge elicited a series of increasingly sophisticated canonical treatments of pairing correlations [33][34][35][36][37][38][39][40][41], based on a simple reduced BCS-Hamiltonian for discrete energy levels, which showed that the crossover is completely smooth, but, interestingly, depends on the parity of the number of electrons on the grain, as pointed out by von Delft et al [21].…”

Superconductivity in Ultrasmall Grains: Introduction to Richardson’s Exact Solution

“…Most previous works [2,3,4,6,7,8,9, 10] studied a half-filled band with fixed width 2ω D of uniformly-spaced levels (i.e. ε j = j d), containing N = 2n + b electrons.…”

“…and a parity parameter introduced by Matveev and Larkin (ML) [6] to characterize the even-odd ground state energy difference,…”

“…studies [2]- [6], the first canonical study was that of Mastellone, Falci and Fazio (MFF) [7], who used Lanczos exact diagonalization (with n ≤ 12) and a scaling argument to probe the crossover regime.…”

“…BCS approach [2,3,4] was applied to a reduced BCS Hamiltonian for uniformly spaced, spin-degenerate levels; it suggested that pairing correlations, as measured by the condensation energy E C , vanish abruptly once d exceeds a critical level spacing d c that depends on the parity (0 or 1) of the number of electrons on the grain, being smaller for odd grains (d c 1 ≃ 0.89∆) than even grains (d c 0 ≃ 3.6∆). A series of more sophisticated canonical approaches (summarized in Section 3 below) confirmed the parity dependence of pairing correlations, but established [6]- [11] that the abrupt vanishing of pairing correlations at d c is an artifact of g.c.…”