Sensitivity of superconducting states to the impurity location in layered materials

The family of multi-layered superconductors derived from the doped topological insulator Bi$_2$Se$_3$ has been found to be unusually robust against non-magnetic disorder. Recent experimental studies have highlighted the fact that the location of impurities could play a critical role for this puzzling robustness. Here we investigate the effects of four different types of impurities, on-site, interstitial, intercalated and polar, on the superconducting critical temperature. We find that different components of the scattering potential are active depending on the impurity configuration and choice of orbitals for the effective low-energy description of the normal state. For the specific case of Bi$_2$Se$_3$-based superconductors, we find that only the symmetric share of impurity configurations contribute to scattering, such that polar impurities are completely inactive. We also find that a more dominant mass-imbalance term in the normal-state Hamiltonian can make the superconducting state more robust to intercalated impurities, in contrast to the case of on-site or interstitial impurities.