Possible two-component pairings in electron-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ based on a tight-binding model

Recent experiments show the spontaneous breaking of rotational symmetry in the superconducting topological insulators $M_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ ($M$ represents Cu, Sr, or Nd), suggesting that the pairing belongs to a two-dimensional representation of the $D_{3d}$ symmetry group of the crystal. Motivated by this progress, we construct an exhaustive list of possible two-component pairings of the $M_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ superconductors, both for the odd-parity $E_u$ representation and for the even-parity $E_g$ representation. Starting from a tight-binding model for the normal phase of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and $M_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$, we firstly construct the pairing channels in the spin-orbital basis, up to second-nearest-neighbor pairing correlations in the basal plane. We then infer the properties of these pairings by transforming them to the band (pseudospin) basis for the conduction band. A comparison with the experiments leads to several multichannel pairings as promising pairings for $M_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ superconductors. Besides a nematic and time-reversal symmetric pairing combination, the other pairings that we have identified are chiral and nematic at the same time, which may be nonunitary and have a spontaneous magnetization. A complementary set of experiments are proposed to identify the true pairing symmetries of these superconductors and their evolution with the doping concentration $x$.