The BCS-Bogoliubov gap equation with external magnetic field and the first-order phase transition

TL;DR

This paper analyzes the BCS-Bogoliubov gap equation under an external magnetic field in type I superconductors, demonstrating a first-order phase transition and deriving explicit expressions for the entropy gap.

Contribution

It introduces a detailed analysis of the gap equation with magnetic field, proving the existence of a critical magnetic field and characterizing the phase transition as first order.

Findings

Existence of a unique critical magnetic field as a smooth function of temperature.

Identification of a first-order phase transition from normal to superconducting state.

Explicit expression derived for the entropy gap.

Abstract

We deal with a type I superconductor in a constant external magnetic field. We obtain the BCS-Bogoliubov gap equation with external magnetic field and apply the implicit function theorem to it. We show that there is a unique magnetic field (the critical magnetic field) given by a smooth function of the temperature and that there is also a unique nonnegative solution (the gap function) given by a smooth function of both the temperature and the external magnetic field. Using the grand potential, we show that the transition from the normal state to the superconducting state in a type I superconductor is of the first order. Moreover we obtain the explicit expression for the entropy gap.