Reduced pair breaking from extended disorder in unconventional superconductors: implications to 4Hb-TaS$_2$
Unconventional superconductivity is generally expected to be strongly suppressed by nonmagnetic disorder, as captured by Abrikosov--Gor'kov (AG) theory. However, several materials, including transition metal dichalcogenides, exhibit signatures of unconventional pairing despite relatively high resistivities, suggesting a breakdown of the conventional relation between momentum relaxation and pair breaking. Here, we study this problem in H-phase transition metal dichalcogenides by computing the disorder-dressed pairing susceptibility. We employ a multiband model with spin-orbit coupling and include an impurity potential that mimics a common lattice defect, namely a chalcogen vacancy or site ad-atom. This yields to an extended impurity potential, which we compare with the commonly considered on-site (point defect) potential. We evaluate the momentum-relaxation rate and the pair-breaking rate on equal footing. We find that extended impurity potentials lead to a parametrically reduced pair-breaking rate compared to the transport scattering rate, with $\Gamma \tau_D \sim 1/3$ over a wide parameter range. This reduction originates from the momentum structure of the disorder potential, which partially matches the internal structure of the superconducting gap and suppresses pair-breaking processes. As a result, unconventional pairing states are significantly more robust than predicted by standard AG theory. Our results provide a natural explanation for the persistence of unconventional superconductivity in systems with strong disorder and substantially alleviate the apparent conflict between high resistivity and unconventional pairing in materials such as 4Hb-TaS$_2$.