Topological superconductivity in two-dimensional $\pi$-junction Dirac semimetals

Odd-parity pairings offer a natural pathway for realizing topological superconductivity. When two identical even-parity superconductors form a $\pi$-junction, the metallic material sandwiched between them experiences an effective odd-parity pairing, facilitating the emergence of topological superconductivity in the intermediate region. In this work, we consider the intermediate material to be a two-dimensional spin-orbit-coupled Dirac semimetal. When the two superconductors are conventional s-wave superconductors, we find that a helical topological superconductor can be realized. This phase is characterized by the presence of a pair of helical Majorana edge states. Interestingly, when the superconductors are $s_{\pm}$-wave superconductors, we observe not only the helical topological superconductor but also an unconventional topological superconductor. The latter is distinguished by the existence of two pairs of helical Majorana edge states, despite the fact that the global topological invariants for this system take on trivial values. By further applying an in-plane magnetic field, we demonstrate that second-order topological superconducting phases can be achieved. These phases host isolated Majorana corner modes as well as twofold Majorana corner modes. Our findings reveal that two-dimensional $\pi$-junction Dirac semimetals can support a rich variety of topological superconducting phases, offering a versatile platform for exploring exotic topological phenomena.