Superconductivity in ultrasmall metallic grains

Several recent papers have predicted parity effects, based on even-odd ground state energy differences, in ultrasmall (nm scale) superconductors having a discrete electronic eigenspectrum with mean level spacing d ̃ \~Δ (bulk gap). The motivation for the present paper is to analyze the measurability of these and related parity effects in the present generation of experiments [e.g., those of Ralph, Black, and Tinkham (RBT)]. To this end we develop a general theory of superconductivity in ultrasmall metallic grains, based on calculating the eigenspectrum using a generalized BCS variational approach. We discuss how conventional mean field theory breaks down with decreasing sample size, how the so-called blocking effect weakens pairing correlations in states with nonzero total spin, and how this affects the discrete eigenspectrum's behavior in a magnetic field, which favors nonzero total spin. Our calculations qualitatively reproduce the magnetic-field-dependent tunneling spectra for individual aluminum grains measured by RBT. Our main results regarding parity effects are (i) the conclusion that those based on even-odd ground state energy differences are currently not measurable and (ii) the proposal of a parity effect for the pair-breaking energy, which should be measurable provided that the grain size can be controlled sufficiently well.