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

We present a systematic study of the electronic specific heat jump ($\Delta C_{\rm el}$)atthesuperconductingtransitiontemperature$T_c$ of K${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$. Both $T_c$ and $\mathrm{\Delta }{C}_{\mathrm{e}\mathrm{l}}$ monotonously decrease with increasing $x$. The specific heat jump scales approximately with a power-law, $\mathrm{\Delta }{C}_{\mathrm{e}\mathrm{l}}\propto T_c^{\beta }$, 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 $\Delta C_{\rm el} \propto T^3$hasbeenfound\mathrm{.}Boththe$T$dependenceof$C_{\rm el}(T)$inthesuperconductingstateandthenearlyquadraticscalingof$\Delta C_{\rm el}$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 $x>0$.