2007/06/16 by Hao, Zhidong
#FOS: Physical sciences #Superconductivity (cond-mat.supr-con)
paper · doi:10.48550/arxiv.0706.2392
This paper concerns the solution of the self-consistency equation for energy gap parameter Δ\bf k in the BCS theory of superconductivity. We show that there exists a well-defined relation between the solution for energy gap parameter amplitude |Δ\bf k| for a general interaction V_\bf k,\bf k' and energy gap Δ obtained by using the cut-off approximation. The relation between |Δ\bf k| and Δ indicates that Δ is a weighted average over |Δ\bf k| of electronic states within cut-off energy ξc around the Fermi surface. In this interpretation for Δ, ξc is not a property of V_\bf k,\bf k', but a parameter specifying the energy range within which the weighted average over |Δ\bf k| is taken. We show that the proper choice for the value of ξc is only a few kBTc (i.e., ξc/kBTc is about 3 or 4). We also show that the cut-off approximation, even with ξc/kBTc=∞, is a good approximation when it is used to calculate quantities such as the condensation energy and the specific heat, but it leads to significant overestimation for the Josephson critical current density of a Josephson junction if ξc/kBTc ≫ 1 is assumed.