Sea of Majorana fermions from pseudo-scalar superconducting order in three dimensional Dirac materials

We find that singlet superconducting pairing can lead to Majorana fermions in three dimensional Dirac superconductors (3DDS) if the pairing order parameter is a pseudo-scalar, i.e. it changes sign under mirror reflection. The pseudo-scalar superconducting order parameter, ${\mathrm{\Delta }}_5$ can close and reopen the spectral gap caused by the scalar Dirac mass $m$ in a three-dimensional Dirac material (3DDM), giving rise to a two-dimensional Majorana sea (2DMS) at the plane of the gap kink. By bringing the Hamiltonian into a canonical form which then gives the winding number, we show that this system belongs to the DIII class of topological superconductors. We calculate the transport signature of 2DMS, namely a perfect Andreev-Klein transmission that manifests in a robust peak in the differential conductance. Further, we find the $4\pi$ periodicity in the ${\mathrm{\Delta }}_5\mathrm{\mid }m\mathrm{\mid }{\mathrm{\Delta }}_5$ Josephson junc- tions. Gauge transformed version of the present scenario implies that the interface of a conventional s-wave superconductor with a peculiar 3DDM whose mass is a pseudo-scalar, m5 also hosts a 2DMS.