Unconventional superconductivity and surface pairing symmetry in half-Heusler compounds

Signatures of nodal line/point superconductivity [Kim et al., Sci. Adv. 4, eaao4513 (2018); Brydon et al., Phys. Rev. Lett. 116, 177001 (2016)] have been observed in half-Heusler compounds, such as LnPtBi (Ln = Y, Lu). Topologically nontrivial band structures, as well as topological surface states, have also been confirmed by angular-resolved photoemission spectroscopy in these compounds [Liu et al., Nat. Commun. 7, 12924 (2016)]. In this paper, we present a systematical classification of possible gap functions of bulk states and surface states in half-Heusler compounds and the corresponding topological properties based on the representations of crystalline symmetry group. Different from all the previous studies based on the four band Luttinger model, our study starts with the six-band Kane model, which involves both four p-orbital type of ${\mathrm{\ensuremath{\Gamma}}}₈$ bands and two s-orbital type of ${\mathrm{\ensuremath{\Gamma}}}₆$ bands. Although the ${\mathrm{\ensuremath{\Gamma}}}₆$ bands are away from the Fermi energy, our results reveal the importance of topological surface states, which originate from the band inversion between ${\mathrm{\ensuremath{\Gamma}}}₆$ and ${\mathrm{\ensuremath{\Gamma}}}₈$ bands, in determining surface properties of these compounds in the superconducting regime by combining topological bulk state picture and nontrivial surface state picture.