Two-band superconductors: Extended Ginzburg-Landau formalism by a systematic expansion in small deviation from the critical temperature

TL;DR

This paper develops an extended Ginzburg-Landau formalism for two-band superconductors using a systematic expansion near the critical temperature, capturing differences in band condensates and providing accurate thermodynamic predictions.

Contribution

It introduces a systematic expansion-based extended GL theory that accounts for differences in band condensates, improving upon previous models.

Findings

The extended GL theory accurately predicts the thermodynamic critical magnetic field.

Numerical results match full BCS calculations across a wide temperature range.

The theory reveals the leading corrections to the standard GL model for two-band superconductors.

Abstract

We derive the extended Ginzburg-Landau (GL) formalism for a clean s-wave two-band superconductor by employing a systematic expansion of the free-energy functional and the corresponding matrix gap equation in powers of the small deviation from the critical temperature tau = 1-T/T_c. The two lowest orders of this expansion produce the equation for T_c and the GL theory. It is shown that in agreement with previous studies, the two-band GL theory maps onto the single-band GL model and thus fails to describe the difference in the spatial profiles of the two band condensates. We prove that except for some very special cases, this difference appears already in the leading correction to the GL theory, which constitutes the extended GL formalism. We derive linear differential equations that determine the leading corrections to the band order parameters and magnetic field, discuss the validity of these equations, and consider examples of an important interplay between the band condensates. Finally, we present numerical results for the thermodynamic critical magnetic field and temperature-dependent band gaps (at zero field), which are in a very good agreement with those obtained from the full BCS approach in a wide temperature range. To this end, we emphasize the advantages of our extended GL theory in comparison with the often used two-component GL-like model based on an unreconstructed two-band generalization of the Gor'kov derivation.