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

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.

3 figures, Keywords: BCS-Bogoliubov gap equation, external magnetic field, FFLO-like state, first-order phase transition, implicit function theorem

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