To study thermodynamical properties of the disorder-induced transition between $s_{ \pm}$ and $s_{ ++}$ superconducting gap functions, we calculate the grand thermodynamic potential $\Omega$ in the normal and the superconducting states. Expression for the difference between the two, $\Delta\Omega$, is derived for a two-band model for Fe-based systems with nonmagnetic impurities. The disorder is considered in a $\mathcal{ T}$-matrix approximation within the multiband Eliashberg theory. In the vicinity of the Born limit near the $s_{ \pm}$-to-$s_{ ++}$ transition, we find two solutions obtained for opposite directions of the system's evolution with respect to the impurity scattering rate. By calculating the change in entropy $\Delta S$ and the change in electronic specific heat $\Delta C$ from $\Delta\Omega$, we show that such a hysteresis is not due to the time-reversal symmetry breaking state, but it rather points out to the first order phase transition induced by the nonmagnetic disorder. Based on the $\Delta\Omega$ calculations, phase diagram is plotted representing the energetically favourable $s_{ \pm}$ and $s_{ ++}$ states and the transition between them. At finite temperature, a first order phase transition line there is limited by a critical end point. Above that point, the sharp $s_{ \pm} \to s_{ ++}$ transition transforms to a crossover between $s_{ \pm}$ and $s_{ ++}$ states.
Categories: cond-mat.supr-con