Parity effect in mesoscopic and nanoscopic superconducting particles

Superconductivity in small metallic specimens is studied with regard to the size dependence of the parity gap (ΔP), a parameter distinguishing between the energy of even and odd number of electrons in the granule. ΔP is shown to be an increasing function of level spacing δ. The energy gap of superconductor Δ, on the other hand, decreases with increasing δ and vanishes at δ = δc which is of the order of Δ. However, non-zero value of ΔP persists above δc in a gapless superconducting-insulating state. Level degeneracy in small specimens having perfect geometry changes the size dependence of the parity gap, the Josephson effect, and flux quantization. Parity gap is evaluated using an interpolation procedure between the continuum limit (δ ≪ Δ), the moderate mesoscopic regime (δ ∼ Δ), and the nanoscopic scale (δ ≫ Δ), for which an exact solution to the pairing problem is provided with the numeric diagonalization of system Hamiltonian in a small metallic cluster