Topological phases and Majorana modes in spin–orbit-coupled superconducting systems with mixed d -wave and extended s -wave pairings

Based on the spin-generalized Bogoliubov–de Gennes theory, we investigate topological phase transitions and Majorana zero modes in spin–orbit-coupled superconducting systems with mixed d-wave and extended s-wave pairings. The phase diagram is established for both pure Rashba and mixed Rashba–Dresselhaus scenarios, revealing that a finite Dresselhaus component induces an intervening first-order topological phase between the trivial and second-order topological phases. The Fermi-surface analysis shows that these topological transitions correspond to the changing number of pairing nodes enclosed by the spin-split Fermi surfaces. In the square-loop geometry, the presence of both inner and outer boundaries enables the realization of both Majorana edge and corner modes, with asymmetric distributions arising from the reduced C2v symmetry in the mixed spin–orbit interaction case. A magnetic flux vortex modulates the number and spatial distribution of the Majorana states, inducing multiple level crossings associated with distinct zero-energy conductance patterns when the system is near the phase boundary, while remaining more robust deep inside the topological phase. Moreover, charge and spin currents exhibit jumps at the level crossings, providing complementary experimental signatures. Our theoretical predictions highlight the mesoscopic square loop as a promising candidate for realizing tunable Majorana states and detecting higher-order topological superconductivity.