What decides whether disorder destroys pairing or only phase coherence?
In plain words
Disorder can end superconductivity in two ways: by breaking the electron pairs, or by leaving the pairs intact but stopping them from moving together in step. Some films, such as amorphous indium oxide and titanium nitride, keep a pairing gap into the insulator, while in others the gap seems to close together with the transition temperature, and what selects the route is not known.
Precise statement
Near the disorder-tuned 2D superconductor-insulator transition, the fermionic route (Coulomb-enhanced suppression of the pairing amplitude, Finkel'stein) has the gap $\Delta\to 0$ together with $T_c$, while the bosonic route keeps a finite $\Delta$ and a pseudogap above $T_c$, with $T_c\to 0$ from phase fluctuations or localization of preformed pairs (Sacepe et al., Nature Physics 2011, in InOx). Identify the material parameter that selects the route (for example $k_F\ell$, the ratio of Coulomb to pairing strength, or the Debye energy relative to the level spacing in a localization volume) and give a criterion that predicts the route for a given film.
What would settle it
Tunneling measurements of $\Delta(T)$ and $T_c$ across disorder series in several material families, matched to a theory whose criterion predicts each family's route in advance.
Status in the literature
Unverified note
A 2020 review describes both routes and granular behavior without a predictive criterion (Sacepe, Feigel'man and Klapwijk, Nature Physics 16, 734, 2020); a first-order disorder-driven transition was reported in InOx in 2025 (Charpentier et al., Nature Physics 21, 104).