arXiv:0910.2352v1 [cond-mat.supr-con] 13 Oct 2009 Nanoscale superconductors: quantum confinement and spatially dependent Hartree-Fock potential Yajiang Chen, M. D. Croitoru, A. A. Shanenko, * and F. M. Peeters † Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium (Dated: October 13, 2009) 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. PACS numbers: 74.78.-w, 74.78.Na I. INTRODUCTION Advances in nanofabrication technology resulted re- cently in high-quality metallic superconducting ultrathin nanofilms 1,2,3 and nanowires. 4,5,6,7 In most samples the electron mean free path was estimated to be about or larger than the nanofilm/nanowire thickness. 2,4,7 In this case the effects of the transverse quantization are not shadowed by impurity scattering and, hence, the con- duction band splits up into a series of single-electron subbands resulting from the quantized transverse modes. This will have a pronounced effect on the superconduct- ing properties (see, for instance, Refs. 8 and 9 and ref- erences therein). Notice that high-quality nanofilms do not exhibit significant indications of defect- or phase- driven suppression of superconductivity (see discussion in Ref. 2). For high-quality nanowires the phase-fluctuation effects were shown to seriously influence the supercon- ducting state only in narrowest aluminum specimens with width ≈ 5 − 8 nm. 4,7,10 Thus, the transverse quantum confinement is the major mechanism governing the su- perconducting properties in this case. Therefore, it is timely to study in a more detail a clean nanoscale super- conductor in the presence of quantum confinement. Quantum confinement breaks the translational sym- metry and, so, the superconducting order parameter be- comes position dependent. The well-known BCS ansatz for the ground state wave function is not applicable in this case, and the Bogoliubov-de Gennes (BdG) equa- tions are a relevant tool to investigate equilibrium super- conducting properties. Recent numerical studies of the BdG equations for nanofilms 8 and nanowires 9,11,12 show that the transverse quantum confinement has a substan- tial impact on the superconducting solution. However, the BdG equations investigated in Refs. 8, 9, 11 and 12, were solved without the Hartree-Fock (HF) potential. The reason is that in bulk, the superconducting solution is not sensitive to the HF term in the BdG equations 13 , and one can assume that a similar conclusion holds for the broken translational symmetry. However, at present there is no detailed investigations on this subject and, so, such a study is needed. In the bulk BdG equations, the HF potential is not spatially dependent and, so, it produces the same contri- bution to all single-electron energies, with no dependence on the relevant quantum numbers. Hence, the Fermi level (the chemical potential) acquires the same contribution, as well, and the single-electron energies measured from the Fermi level are not changed. It is well-known that the BdG equations are derived within the grand canoni- cal formalism and, so, the electron energies appearing in the basic expressions absorb the chemical potential. As a result, the superconducting solution is insensitive to the HF potential. The situation is different in the presence of quantum confinement. The translational symmetry is now broken, the HF mean field is position dependent, and, so, its contribution to the single-electron energies is a function of the relevant quantum numbers. Fur- thermore, the single-electron wave functions themselves are influenced by the presence of the HF field, i.e., an additional spatially-dependent potential. Therefore, one can expect that the HF term in the BdG equations can change the superconducting solution in the presence of quantum confinement. It is of importance to clarify to what extent this will be through. In particular, this con-