Applicability of Bardeen–Cooper–Schrieffer theory to small-sized superconductors: Role of Cooper-pair binding energy

• We analyze the criterion of applicability of BCS theory to small-sized superconductors.
• We focus on the ground state interaction energy.
• BCS result for this quantity is accurate provided that level spacing is larger than pair binding energy.
• This quantity can be much smaller than the gap.

We analyze conditions of applicability of grand-canonical mean-field Bardeen–Cooper–Schrieffer theory to the evaluation of an interaction energy in the ground state of small-sized superconductors. We argue that this theory fails to describe correctly an interaction energy, when an average distance between energy levels near the Fermi energy due to the size quantization becomes of the order of the single-pair binding energy. In conventional superconductors, this quantity is much smaller than the superconducting gap.