Can electron-phonon coupling mediated superconductivity coexist within the topologically protected edge states?

Topological superconductivity is an intriguing quantum phenomenon that may potentially host an elementary block of fault-tolerant quantum computing—the Majorana fermion. This phenomenon is a combination of two quantum effects: the topologically protected edge states and superconductivity. In two-dimensional topological insulators, the topological edge states are metallic states, while the surfaces are insulating. As the superconductivity requires metallic states, many recent studies reported that the superconducting states may coexist within the edge states induced by proximity to the bulk. However, the edge states are topologically protected, and it remains unknown whether Cooper-pair formation is compatible with spin-angular momentum locking within the edge states. Here we will discuss the potential barrier to introduce intrinsic superconductivity within the edge states. Particularly, we will delve into the superconductivity within the edge states of a two-dimensional topological insulator mediated by electron-phonon coupling. This study demonstrates that the later form of superconductivity is only possible outside the edge states.