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Energy Gap Evolution Across the Superconductivity Dome in Single Crystals of (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$

The mechanism of unconventional superconductivity in iron-based superconductors (IBSs) is one of the most intriguing questions in current materials research. Among non-oxide IBSs, (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ has been intensively studied because of its high superconducting transition temperature and fascinating evolution of the superconducting gap structure from being fully isotropic at optimal doping (\(x\approx\)0.4) to becoming nodal at $x > $0.8. Although this marked evolution was identified in several independent experiments, there are no details of the gap evolution to date because of the lack of high-quality single crystals covering the entire K-doping range of the superconducting dome. We conducted a systematic study of the London penetration depth, $λ(T)$, across the full phase diagram for different concentrations of point-like defects introduced by 2.5 MeV electron irradiation. Fitting the low-temperature variation with the power law, $Δλ\sim T^{n}$, we find that the exponent $n$ is the highest and $T_c$ suppression rate with disorder is the smallest at optimal doping, and they evolve with doping being away from optimal, which is consistent with increasing gap anisotropy, including an abrupt change around $x\simeq 0.8$, indicating the onset of nodal behavior. Our analysis using a self-consistent $t$-matrix approach suggests the ubiquitous and robust nature of s$_{\pm}$ pairing in IBSs and argues against a previously suggested transition to a $d-$wave state near $x=1$ in this system.