The effect of hydrostatic pressure and partial Na substitution on the normal-state properties and the superconducting transition temperature ($T_c$) of K$_{1-x}$Na$_x$Fe$_2$As$_2$ single crystals were investigated. It was found that a partial Na subs
titution leads to a deviation from the standard $T^2$ Fermi-liquid behavior in the temperature dependence of the normal-state resistivity. It was demonstrated that non-Fermi liquid like behavior of the resistivity for K$_{1-x}$Na$_{x}$Fe$_2$As$_2$ and some KFe$_2$As$_2$ samples can be explained by disorder effect in the multiband system with rather different quasiparticle effective masses. Concerning the superconducting state our data support the presence of a shallow minimum around 2 GPa in the pressure dependence of $T_c$ for stoichiometric KFe$_2$As$_2$. The analysis of $T_c$ in the K$_{1-x}$Na$_{x}$Fe$_2$As$_2$ at pressures below 1.5 GPa showed, that the reduction of $T_c$ with Na substitution follows the Abrikosov-Gorkov law with the critical temperature $T_{c0}$ of the clean system (without pair-breaking) which linearly depends on the pressure. Our observations, also, suggest that $T_c$ of K$_{1-x}$Na$_x$Fe$_2$As$_2$ is nearly independent of the lattice compression produced by the Na substitution. Further, we theoretically analyzed the behavior of the band structure under pressure within the generalized gradient approximation (GGA). A qualitative agreement between the calculated and the recently in de Haas-van Alphen experiments [T. Terashima et al., Phys.Rev.B89, 134520(2014)] measured pressure dependencies of the Fermi-surface cross-sections has been found. These calculations, also, indicate that the observed minimum around 2~GPa in the pressure dependence of $T_c$ may occur without a change of the pairing symmetry.
We present a systematic study of the electronic specific heat jump ($Delta C_{rm el}$) at the superconducting transition temperature $T_c$ of K$_{1-x}$Na$_x$Fe$_2$As$_2$. Both $T_c$ and $Delta C_{rm el}$ monotonously decrease with increasing $x$. The
specific heat jump scales approximately with a power-law, $Delta C_{rm el} propto T_c^{beta}$, with $beta approx 2$ determined by the impurity scattering rate, in contrast to most iron-pnictide superconductors, where the remarkable Budko-Ni-Canfield (BNC) scaling $Delta C_{rm el} propto T^3$ has been found. Both the $T$ dependence of $C_{rm el}(T)$ in the superconducting state and the nearly quadratic scaling of $Delta C_{rm el}$ at $T_c$ are well described by the Eliashberg-theory for a two-band $d$-wave superconductor with weak pair-breaking due to nonmagnetic impurities. The disorder induced by the Na substitution significantly suppresses the small gaps leading to gapless states in the slightly disordered superconductor, which results in a large observed residual Sommerfeld coefficient in the superconducting state for $x > 0$.
