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We investigate the phase diagram of dirty two-band superconductors. Thispaper primarily focuses on the properties and observability of thetime-reversal symmetry-breaking $s+is$ superconducting states, which can begenerated in two-band superconductors by interband impurity scattering. We showthat such states can appear in two distinct ways. First, according to apreviously discussed scenario, the $s+is$ state can form as an intermediatephase at the impurity-driven crossover between $s_{\pm}$ and $s_{++}$ states.We show that there is a second scenario where domains of the $s+is$ stateexists in the form of an isolated dome inside the $s_{\pm}$ domain, completelydetached from the transition between $s_{\pm}$ and $s_{++}$ states. Wedemonstrate that in both cases the $s+is$ state, generated by impurityscattering exists in an extremely small interval of impurity concentrations.Although this likely precludes direct experimental observation of the $s+is$state formation due to this mechanism, this physics leads to the appearance ofa region inside both the $s_{\pm}$ and $s_{++}$ domains with unusual propertiesdue to softening of normal modes.

Phase diagram of dirty two-band superconductors and observability of impurity-induced $s+is$ state