Superconductivity in Two-Dimensional Systems with Unconventional Rashba Bands

In two-dimensional system with Rashba spin-orbit coupling, it is well-known that superconductivity can have mixed spin-singlet and -triplet parity, and the $\boldsymbol{d}$-vector of spin-triplet pairing is parallel to $\boldsymbol{g}$-vector of Rashba spin-orbit coupling. Here, we propose a model to describe a two-dimensional system with unconventional Rashba bands and study its superconductivity. We show that the $\boldsymbol{d}$-vector of spin-triplet pairing can be either parallel or perpendicular to $\boldsymbol{g}$-vector of Rashba spin-orbit coupling depending on the different pairing interaction. We also propose a junction to generate tunneling current depending on the direction of $\boldsymbol{d}$-vector. It provides a detectable evidence to distinguish these two different but very similar pairing channels. Furthermore, we find this model can give arise to a subleading spin-singlet chiral $p$-wave topological superconducting state. More significantly, we find that such unconventional Rashba bands and unconventional superconudcting pairings can be realized on surface of some superconducting topological materials, such as trigonal layered PtBi$_{2}$.