Superconductivity in ultrasmall metallic grains

We review recent experimental and theoretical work on superconductivity in ultrasmall metallic grains, i.e, grains sufficiently small that the conduct ion electron energy spectrum becomes discrete. The discrete excitation spec trum of an individual grain can be measured by the technique of single-elec tron tunneling spectroscopy, and reveals parity effects indicative of pairi ng correlations in the grain. After introducing the discrete BCS model that has been used to model such grains, we review a phenomenological, grand-ca nonical, variational BCS theory describing the paramagnetic breakdown of th ese pairing correlations with increasing magnetic field. We also review rec ent canonical theories that have been developed to describe how pairing cor relations change during the crossover, with decreasing grain size, from the -bulk limit to the limit of few electrons, and compare their results to tho se obtained using Richardson's exact solution of the discrete BCS model.