Superconducting specific heat jump Δ C ∝ T_c^β (β ∝ 2) and gapless Fermi surfaces in K_1-xNa_xFe₂As₂

We present a systematic study of the electronic specific heat jump (Δ C_el) at the superconducting transition temperature T_c of K_1-xNa_xFe₂As₂. Both T_c and Δ C_el monotonously decrease with increasing x. The jump scales approximately with a novel power-law: Δ C_el ∝ T_c^β with β ∼ 2 determined by the impurity scattering rate. This finding is in sharp contrast to most of all iron-pnictide superconductors with a cubic Bud'ko-Ni-Canfield (BNC) scaling. Our observations, also, suggests that disorder diminishes the small gaps leading to partial gapless superconductivity which results in a large residual Sommerfeld coefficient in the superconducting state for x > 0. Both T-dependence of C_el(T) in the superconducting state and the nearly quadratic scaling of Δ C_el at T_c are well described by the Eliashberg-theory for a single-band d-wave superconductor with weak pair-breaking due to nonmagnetic impurities having reduced the density of superconducting quasi-particles.