Tri-Dirac surface modes in topological superconductors

We propose a new type of topological surface modes having cubic dispersion in three-dimensional topological superconductors. Lower order dispersions are prohibited by the threefold rotational symmetry and time-reversal symmetry. Cooper pairing in the bulk changes sign under improper rotations, akin to ${}^3\mathrm{He}$-B. The surface manifestations are a divergent surface density of states at the Fermi level and isospins that rotate three times as they circle the origin in momentum space. We propose that Heusler alloys with band inversion are candidate materials to harbor the novel topological superconductivity.

Figure 1

(a) The energy dispersion of the Bogoliubov excitations around a tri-Dirac point. (b) The isoenergy contours and the pseudospin structure of $h_{3/2}\left(\mathbf{k}\right)$ along the contour taking $c_2=2c_3=1$.

Figure 2

Three possible scenarios for the Fermi level crossing the bulk ${\Gamma }_8$ bands and the schematic dispersions of the surface states [from left to right: case (ii), (i), and (iii)]. Solid and dotted lines mean that the corresponding band has positive and negative pairing amplitude, respectively, in the nontrivial superconducting phase. See Ref. [41] for a discussion of the case of degenerate bands in the presence of inversion.

Figure 3

(a) Schematic plot of the differential conductance on the surface of TSCs with Dirac (blue) and tri-Dirac surface states inside the bulk superconducting gap. (b) Typical symmetry-forbidden scattering vectors on the isoenergy contour of the tri-Dirac surface states. Black arrows indicate scattering vectors forbidden by TRS, and red arrows are vectors strongly suppressed due to the special pseudospin structure of tri-Dirac surface states, also plotted on the contour.