Superconducting nanowires: quantum confinement and spatially dependent Hartree-Fock potential

It is well known that, in bulk, the solution of the Bogoliubov-de Gennes equations is the same whether or not the Hartree-Fock term is included. Here the Hartree-Fock potential is position independent and so gives the same contribution to both the single-electron energies and the Fermi level (the chemical potential). Thus, the single-electron energies measured from the Fermi level (they control the solution) stay the same. This is not the case for nanostructured superconductors, where quantum confinement breaks the translational symmetry and results in a position-dependent Hartree-Fock potential. In this case its contribution to the single-electron energies depends on the relevant quantum numbers. We numerically solved the Bogoliubov-de Gennes equations with the Hartree-Fock term for a clean superconducting nanocylinder and found a shift of the curve representing the thickness-dependent oscillations of the critical superconducting temperature to larger diameters.