Parity-Affected Superconductivity in Ultrasmall Metallic Grains

Abstract

We investigate the breakdown of BCS superconductivity in {\em ultra}\/small 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 $\Delta_P (d,T)$. Both the $T=0$ critical level spacing $d_{c,P} $ and the critical temperature $T_{c,P} (d)$ at which $\Delta_P = 0$ 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.Comment: 4 pages RevTeX, 1 encapsulated postscript figure, submitted to Physical Review Letter

Visualizing superconductivity in FeSe nanoflakes onSrTiO3by scanning tunneling microscopy

“…Superconductivity in reduced dimensions, including ultrathin films, 1-3 nanowires 4 and nanograins, [5][6][7][8][9][10] has still been a hot topic of interest due to its potential for developing dissipationless nanoelectronics. However, as the size of a superconductor is shrunk to nanoscale, quantum size effects (QSE) would become important and can dramatically change its superconducting properties.…”

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.…”

“…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.…”

“…Recent experiments by Ralph, Black and Tinkham, involving the observation of a spectroscopic gap indicative of pairing correlations in ultrasmall Al grains [1], have inspired a number of theoretical [2]- [11] studies of how superconducting pairing correlations in such grains are affected by reducing the grains' size, or equivalently by increasing its mean level spacing d ∝ Vol −1 until it exceeds the bulk gap ∆. In the earliest of these, a grand-canonical (g.c.)…”

“…In the earliest of these, a grand-canonical (g.c.) 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.…”