Parity-Affected Superconductivity in Ultrasmall Metallic Grains

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 ( $P=\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}/\mathrm{o}\mathrm{d}\mathrm{d}$) of the number of electrons on the island. Assuming equally spaced levels, we solve the parity-dependent BCS gap equation for the pairing parameter ${\Delta }_P(d,T)$. The $T=0\mathrm{}$ critical level spacing $d_{c,P}$, the critical temperature $T_{c,P}\left(d\right)$ (at which ${\Delta }_P=0\mathrm{}$), and the condensation energy $E_P$ are parity dependent, and all are so much smaller in the odd than the even case that this should manifest itself in current experiments.