Superfluid density of the $\pm$s-wave state for the iron-based superconductors

We study the superfluid density of the $\pm$s-wave state of the minimal two band model for the Fe-based superconductors and its evolution with impurity concentration. We show that the impurity scattering of the strong coupling limit induces the selfenergy of a generic form $Im \Sigma_{imp} (\omega) \approx i \gamma + i \beta \omega$ beyond a critical impurity concentration $\Gamma_{imp} > \Gamma_{crit}$. This form of $Im \Sigma_{imp} (\omega)$ causes the temperature dependence of the superfluid density $[\rho_{s}(T) - \rho_{s}(0)] \approx - \gamma T^2 -\beta T^3$. Combining with the full gap behavior of $\rho_{s}(T)$ for lower impurity concentration $\Gamma_{imp} < \Gamma_{crit}$, the $\pm$s-wave state produces a continuous evolution of $\Delta \lambda(T)$: exponentially flat $\to T^3 \to T^2$ with increasing impurity concentration that is consistent with the measurements of the Fe pnictide superconductors such as $M$-1111 ($M$=La,Nd,Sm,Pr) and Ba-122 with various dopings, except LaFePO which shows $\Delta \lambda(T) \propto T^{1.2}$ at low temperatures by recent experiment. Our results also demonstrate that the density of states (DOS) measured by thermodynamic properties and the DOS measured by transport properties can in general be different.