Superconductivity in two-dimensional systems with unconventional Rashba bands

In two-dimensional systems with Rashba spin-orbit coupling, it is well known that superconductivity can exhibit mixed spin-singlet and spin-triplet parity, with the $\mathbit{d}$ vector of spin-triplet pairing aligning parallel to the $\mathbit{g}$ vector of Rashba spin-orbit coupling. In this work, we propose a model to describe a two-dimensional system with unconventional Rashba bands and investigate its superconducting properties. We demonstrate that the $\mathbit{d}$ vector of spin-triplet pairing can be either parallel or perpendicular to the $\mathbit{g}$ vector of Rashba spin-orbit coupling, depending on the nature of the pairing interaction. We also propose a junction setup to identify the predominant pairing in such two similar pairing channels. Additionally, our model predicts the emergence of a subleading spin-singlet chiral $p$-wave topological superconducting state. Significantly, we find that such unconventional Rashba bands and superconducting pairings can be realized on the surface of certain superconducting topological materials, such as trigonal layered ${\mathrm{PtBi}}_{2}$.