Superconducting specific heat jump $\Delta C_{\rm el} \propto T_c^\beta \ (\beta \approx 2)$ for K$_{1-x}$Na$_x$Fe$_2$As$_2$

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 Bud'ko-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 $x > 0$.