Nanoscale superconductors: 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. In this case 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 energy measured from the Fermi level (it controls the solution) stays the same. It is not the case for nanostructured superconductors, where quantum confinement breaks the translational symmetry and results in a position dependent Hartree-Fock potential. Now the contribution of the Hartree-Fock mean field to the single-electron energies depends on the relevant quantum numbers. Moreover, the single-electron wave functions can be influenced by the presence of this additional spatially dependent field. We numerically solved the Bogoliubov-de Gennes equations with the Hartree-Fock term for a clean metallic nanocylinder and found a shift of the curve representing the thickness-dependent oscillations of the critical temperature (the energy gap, the order parameter etc.) to larger diameters. Though the difference between the superconducting solutions with and without the Hartree-Fock interaction can, for some diameters, be very significant, the above mentioned shift is less than typical metallic unit-cell dimensions and, so, has no practical worth. This allows one to significantly simplify the problem and, similar to bulk, ignore the Hartree-Fock potential when solving the Bogoliubov-de Gennes equations in the nano-regime.