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

TL;DR

This paper investigates how quantum confinement in superconducting nanowires causes a spatially dependent Hartree-Fock potential, affecting their electronic properties and critical temperature oscillations.

Contribution

It introduces a numerical approach to solve Bogoliubov-de Gennes equations with a position-dependent Hartree-Fock potential in nanostructured superconductors.

Findings

Hartree-Fock potential varies with position in nanostructures.

Quantum confinement shifts critical temperature oscillations.

Critical temperature curves move to larger diameters.

Abstract

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. It 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.