  Recent experiments have revealed the evidence of nodal-line superconductivity
in half-Heusler superconductors, e.g. YPtBi. Theories have suggested the
topological nature of such nodal-line superconductivity and proposed the
existence of surface Majorana flat bands on the (111) surface of half-Heusler
superconductors. Due to the divergent density of states of the surface Majorana
flat bands, the surface order parameter and the surface impurity play essential
roles in determining the surface properties. In this work, we studied the
effect of the surface order parameter and the surface impurity on the surface
Majorana flat bands of half-Heusler superconductors based on the Luttinger
model. To be specific, we consider the topological nodal-line superconducting
phase induced by the singlet-quintet pairing mixing, classify all the possible
translationally invariant order parameters for the surface states according to
irreducible representations of $C_{3v}$ point group, and demonstrate that any
energetically favorable order parameter needs to break time-reversal symmetry.
We further discuss the energy splitting in the energy spectrum of surface
Majorana flat bands induced by different order parameters and non-magnetic or
magnetic impurities. We proposed that the splitting in the energy spectrum can
serve as the fingerprint of the pairing symmetry and mean-field order
parameters. Our theoretical prediction can be examined in the future scanning
tunneling microscopy experiments.
