Thermodynamics of Two - Band Superconductors: The Case of MgB$_{2}$

TL;DR

This paper compares the simple two-band BCS-like $ extit{ extalpha}$-model with full Eliashberg theory for MgB$_{2}$, confirming the $ extalpha$-model's reliability in analyzing thermodynamic properties despite interband scattering.

Contribution

It provides the first theoretical validation of using the $ extalpha$-model for thermodynamic analysis of two-band superconductors like MgB$_{2}$, including effects of interband scattering.

Findings

Good agreement between $ extalpha$-model and Eliashberg calculations.

Only the sum of band contributions has physical meaning for thermodynamic potential.

$ extalpha$-model is validated as a reliable analysis tool.

Abstract

Thermodynamic properties of the multiband superconductor MgB$_{2}$ have often been described using a simple sum of the standard BCS expressions corresponding to $\sigma$- and $\pi$-bands. Although, it is \textit{a priori} not clear if this approach is working always adequately, in particular in cases of strong interband scattering. Here we compare the often used approach of a sum of two independent bands using BCS-like $\alpha$-model expressions for the specific heat, entropy and free energy to the solution of the full Eliashberg equations. The superconducting energy gaps, the free energy, the entropy and the heat capacity for varying interband scattering rates are calculated within the framework of two-band Eliashberg theory. We obtain good agreement between the phenomenological two-band $\alpha$-model with the Eliashberg results, which delivers for the first time the theoretical verification to use the $\alpha$-model as a useful tool for a reliable analysis of heat capacity data. For the thermodynamic potential and the entropy we demonstrate that only the sum over the contributions of the two bands has physical meaning.