Mixed parity octupolar pairing and corner Majorana modes in three dimensions

We identify time-reversal symmetry breaking mixed parity superconducting states that feature eight Majorana corner modes in properly cleaved three-dimensional cubic crystals. As we show,  when an odd-parity isotropic p-wave pairing coexists with cubic symmetry preserving even-parity octupolar d_x²-y² +i d_3z²-r² pairing, the gapless surface Majorana modes of the former get localized at the eight corners, thus yielding an intrinsic third-order topological superconductor (TOTSC). A cousin d_xy +i d_3z²-r² pairing also accommodating eight corner Majorana modes, by virtue of breaking the cubic symmetry, in contrast, yields an extrinsic  TOTSC. We identify doped octupolar (topological or trivial) Dirac insulator as the suitable platform to sustain such unconventional superconductors. In particular, our analysis suggests that the doped octupolar Dirac insulator does not need to be topological to accommodate TOTSC. Furthermore, a simple intraunit cell pairing is energetically most favorable topological  pairing in this system. Finally, we argue that the proposed TOTSC can be experimentally realizable in NaCl, Ti₄XTe₃, with X=Pb, Sn, and other structurally similar compounds under high pressure.