Thermodynamics of Two-Band Superconductors: The Case of MgB2

Thermodynamic properties of the multiband superconductor MgB2 have often been described using a simple sum of the standard BCS expressions corresponding to sigma- and pi bands. However, it is a priori not clear if this approach is working always adequately, particularly 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 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.