Superconducting nanowires: quantum confinement and spatially dependent Hartree–Fockpotential

Abstract

It is well known that, in bulk, the solution of the Bogoliubov–de Gennes equations is thesame whether or not the Hartree–Fock term is included. Here the Hartree–Fock potential isposition independent and so gives the same contribution to both the single-electronenergies and the Fermi level (the chemical potential). Thus, the single-electron energiesmeasured from the Fermi level (they control the solution) stay the same. This is not thecase for nanostructured superconductors, where quantum confinement breaks thetranslational symmetry and results in a position-dependent Hartree–Fock potential. In thiscase its contribution to the single-electron energies depends on the relevant quantumnumbers. We numerically solved the Bogoliubov–de Gennes equations with theHartree–Fock term for a clean superconducting nanocylinder and found a shift of the curverepresenting the thickness-dependent oscillations of the critical superconductingtemperature to larger diameters.