Number theory, periodic orbits, and superconductivity in nanocubes

Abstract

We study superconductivity in isolated superconducting nanocubes and nanosquares of size L in the limit of negligible disorder δ/Δ0≪1 and kFL≫1 for which mean-field theory and semiclassical techniques are applicable, with kF the Fermi wave vector, δ the mean level spacing, and Δ0 the bulk gap. By using periodic orbit theory and number theory we find explicit analytical expressions for the size dependence of the superconducting order parameter. Our formalism takes into account contributions from both the spectral density and the interaction matrix elements in a basis of one-body eigenstates. The leading size dependence of the energy gap in three dimensions seems to be universal as it agrees with the result for chaotic grains. In the region of parameters corresponding to conventional metallic superconductors, and for sizes L≳10 nm, the contribution to the superconducting gap from the matrix elements is substantial (∼20%). Deviations from the bulk limit are still clearly observed even for comparatively large grains L∼50 nm. These analytical results are in excellent agreement with the numerical solution of the mean-field gap equation.

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