Observation of a ubiquitous three-dimensional superconducting gap function in optimally-doped Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$

The knowledge of the quasi-three-dimensional (3D) superconducting (SC) gap is essential for understanding the superconducting mechanism of the iron-pnictides highlighted by their multiband and quasi-3D electronic structure. By using the $k_z$-capability of angle-resolved photoemission, we completely determined the SC gap on all five FSs in 3D on Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ samples. We found a significant $k_z$ dispersion of the SC gap which can only derive from interlayer pairing. Remarkably, the SC energy gaps can be described by a single 3D gap function with two energy scales characterizing the strengths of intra-layer ($\Delta_1$) and interlayer ($\Delta_2$) pairing. The anisotropy ratio $\Delta_2/\Delta_1$, determined from the gap function, is close to the $c$-axis anisotropy ratio of the magnetic exchange coupling $J_c/J_{ab}$ in the parent compound. The ubiquitous gap function for all the 3D FSs reveals that pairing is short-ranged and strongly constrain the possible pairing force in the pnictides.