  Recently, unconventional superconductivity having a zero-bias conductance
peak is reported in doped topological Dirac semimetal (DSM) with lattice
distortion. Motivated by the experiments, we theoretically study the possible
symmetry-lowering lattice distortions and their effects on the emergence of
unconventional superconductivity in doped topological DSM. We find four types
of symmetry-lowering lattice distortions that reproduce the crystal symmetries
relevant to experiments from the group-theoretical analysis. Considering
inter-orbital and intra-orbital electron density-density interactions, we
calculate superconducting phase diagrams. We find that the lattice distortions
can induce unconventional superconductivity hosting gapless surface Andreev
bound states (SABS). Depending on the lattice distortions and superconducting
pairing interactions, the unconventional inversion-odd-parity superconductivity
can be either topological nodal superconductivity hosting a flat SABS or
topological crystalline superconductivity hosting a gapless SABS. Remarkably,
the lattice distortions increase the superconducting critical temperature,
which is consistent with the experiments. Our work opens a pathway to explore
and control pressure-induced topological superconductivity in doped topological
semimetals.
