  The robustness against local perturbations, as long as the symmetry of the
system is preserved, is a distinctive feature of topological quantum states.
Magnetic impurities and defects break time-reversal invariance and,
consequently, time-reversal invariant (TRI) topological superconductors are
fragile against this type of disorder. Non-magnetic impurities, however,
preserve time-reversal symmetry and one naively expects a TRI topological
superconductor to persist in the presence of non-magnetic impurities. In this
work, we study the effect of non-magnetic disorder on a TRI topological
superconductor with extended $s$-wave pairing, which can be engineered at the
interface of an Fe-based superconductor and a strongly spin-orbit coupled
Rashba layer. We model two different types of non-magnetic random disorder and
analyze both the bulk density of states and edge state spectrum. Contrary to
naive expectations, we find that the disorder strongly affects the topological
phase by closing the energy gap, while trivial superconducting phases remain
stable and fully gapped. The disorder phase diagram reveals a strong expansion
of a nodal phase with increasing disorder. We further show the decay of the
helical Majorana edge states in the topological phase and how they eventually
disappear with increasing disorder. These results alter our understanding of
effects of impurities and disorder on TRI topological phases and may help
explain the difficulty of experimental observation of TRI topological
superconductors.
