New interpretation for energy gap $\Delta$ of the cut-off approximation in the BCS theory of superconductivity

TL;DR

This paper reinterprets the energy gap parameter in BCS superconductivity, showing it as a weighted average over states within a specific energy range, and clarifies the implications of the cut-off approximation on various physical quantities.

Contribution

It provides a new interpretation of the energy gap  in BCS theory, relating it to a weighted average over electronic states within a defined energy cutoff.

Findings

 is a weighted average over states within c around the Fermi surface

Proper c is a few times k_B T_c, about 3-4 times

Cut-off approximation is accurate for condensation energy and specific heat, but overestimates Josephson current

Abstract

This paper concerns the solution of the self-consistency equation for energy gap parameter $\Delta_{\bf k}$ in the BCS theory of superconductivity. We show that there exists a well-defined relation between the solution for energy gap parameter amplitude $|\Delta_{\bf k}|$ for a general interaction $V_{{\bf k},{\bf k}'}$ and energy gap $\Delta$ obtained by using the cut-off approximation. The relation between $|\Delta_{\bf k}|$ and $\Delta$ indicates that $\Delta$ is a weighted average over $|\Delta_{\bf k}|$ of electronic states within cut-off energy $\xi_c$ around the Fermi surface. In this interpretation for $\Delta$, $\xi_c$ is not a property of $V_{{\bf k},{\bf k}'}$, but a parameter specifying the energy range within which the weighted average over $|\Delta_{\bf k}|$ is taken. We show that the proper choice for the value of $\xi_c$ is only a few $k_BT_c$ (i.e., $\xi_c/k_BT_c$ is about 3 or 4). We also show that the cut-off approximation, even with $\xi_c/k_BT_c=\infty$, 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 $\xi_c/k_BT_c \gg 1$ is assumed.