Superfluid density and specific heat within self-consistent scheme for two-band superconductor

The two gaps in a two-band clean s-wave superconductor are evaluated self-consistently within the quasiclassical Eilenberger weak-coupling formalism with two in-band and one inter-band pairing potentials. Superfluid density, free energy and specific heat are given in the form amenable for fitting the experimental data. Well-known two-band MgB$_2$ and V$_3$Si superconductors are used to test the developed approach. The pairing potentials obtained from the fit of the superfluid density data in MgB$_2$ crystal were used to calculate temperature-dependent specific heat, $C(T)$. The calculated $C(T)$ compares very well with the experimental data. Advantages and validity of this, what we call the "$\gamma$-model", are discussed and compared with the commonly used empirical (and \textit{not self-consistent}) "$\alpha$-model". Correlation between the sign of the inter-band coupling and the signs of the two order parameters is discussed. Suppression of the critical temperature by the inter-band scattering is evaluated and shown to be severe for the inter-band repulsion as compared to the attraction. The data on a strong $T_c$ suppression in MgB$_2$ crystals by impurities suggest that the order parameters on two effective bands of this material may have opposite signs, i.e., may have the $s_{\pm}$ structure similar to the current proposals in iron-based pnictide superconductors.